As I noted in a comment above, I would be avoid using either expression because both are based on outdated and potentially offensive stereotypes of gender roles. In his answer, David M provides a nicely nuanced account of the issues involved for native and nonnative English speakers alike.
Historical Antecedents of "Wears the Pants" and "Wears the Skirt"
The earliest instances in English that use of reversed clothing choices as a metaphor for reversed gender roles involves not pants and skirts, but breeches and petticoats. The playwright Richard Brome wrote a comedy circa 1629–1632 titled The City Wit, or, the Woman Wears the Breaches. Wkipedia's article on the play asserts that the alternate title "the Woman Wears the Breeches" refers to a male character who disguises himself as a woman but continues to wear trousers beneath his skirts. However, the play also features "a shrew and harridan"—the mother-in-law of the main character—and her browbeaten husband; so the phrase "the woman wears the breeches" may be a double entendre if the later meaning of the phrase already existed in 1632.
That meaning clearly existed 20 years later, as the next-earliest reference in a Google Books search establishes—an epigram in John Mennes, Recreation for Ingenious Head-Peeces. Or, A Pleasant Grove for their Wits to walk in (1650):
Megge lets her husband boast of rule and riches,
But she rules all the roast [sic], and wears the breeches.
The next occurrence appears as a comment attached to a (rather impenetrable) proverb in James Howell, Lexicon Tetraglotton, an English-French-Italian-Spanish Dictionary ... With Another Volume of the Choicest Proverbs (1660):
Go dig at Mavorn hill ; Spoken of one whose wife wears the breeches .
Richard Saunders, Saunders Physiognomie, and Chiromancie, Metoposcopie, the Symmetrical Proportions and Signal Moles of the Body (1671) purports to have discovered a physiological indicator of uxoriousness:
Observe the finger of Mercury, or the little finger, if the end thereof exceed the last joint of the Annular, or Ring-finger, such a man Rules in his House, and hath his wife pleasing and obedient to him; but if it be short, and reach not the joynt, that man hath a Shrew, an imperious commanding' woman, that wears the Breeches ; if one hand differ from the other (as it may do) having in on the little finger exceeding the joint, in the other shorter, then it denotes one Wife a Shrew, the other courteous ; and you may know how to distinguish by observing the hands ; for if that hand that shews the lines most conspicuous, have the little finger long, passing the joint of the Annular, then the first Wife is good : if that hand have the shorter finger, then the first Wife is a Shrew, and so of the other.
And John Dunton, Athenian Sport: Or, Two Thousand Paradoxes Merrily Argued to Amuse and Divert the Age (1707) volunteers this gloss on what it means for a woman to wear the breeches:
Paradox XL. Married Women are Men by Conquest ; or, a Paradox proving a true Wife wears the Breeches.
I don't mean that she always wears the Breeches, but that she is so consummately perverse that there's no manner of way to work upon her : A Tiger may be tam'd, a Lion may have his Teeth knock'd out and Claws par'd, and any other sort of Viper its Sting pull'd out ; but do all this to a marry'd Woman, 'twill so provoke herm she'll still act the Man (I mean wear the Breeches in spight of your Teeth).
"Wears the petticoat" as a counterpart to "wears the breeches" first appears in Joseph Addison, The Spectator, no. 482, (September 12, 1711), in the guise of a letter to the paper:
You have given us a lively picture of that kind of husband who comes under the denomination of the Henpeck'd ; but I do not remember that you have ever touched upon one that is of the quite different character, and who, in several places of England, goes by the name of a Cot-quean. I have the misfortune to be joined for life with one of this character, who in reality is more a woman than I am. He was bred up under the tuition of a tender mother, till she had made him as good a housewife as her self. ... Since you have given us the character of a wife who wears the breeches, pray say something of a husband that wears the petticoat. Why should not a female character be as ridiculous in a man, as a male character in one of our sex?
The earliest occurrence in a Google Books search of "wears the trousers" (rather than breeches) in reference to a dominating wife appears in a joke published in Life magazine (September 19, 1889):
Waker: Say, Meeks, how did you ever pluck up courage enough to propose to your wife?
Meeks (whose wife wears the trousers): Why, I didn't.
And the earliest relevant Google Books result for "wears the pants" in that sense occurs in J. S. Robinson, "Women Who Seem Men," in the [Salt Lake City] Young Women's Journal ("Organ of the Y[oung] L[adies'] M[utual] I[mprovement] Associations") (December 1895):
"She wears the pants," you often hear,
And if you look around
A number of such women queer,
May anywhere be found.
They are the bosses of the place,
And manage all affairs,
While dear their husbands, meek of face,
But serve them everywhere.
Louis Menand, Miscellaneous Documents on Divers Subjects (1896) says that the saying about pants and petticoats was originally French:
A young man, I understood to be a Welshman, once came to me to ask me for work. I answered affirmatively that I would give him work. He made the remark that he was a married man and should like to board himself if he could get a house close by. I showed him one he could get cheap if it suited him. He wanted to see it and one day later he came and told me that the house suited him, but he did not like it. I wonder who could be He. I knew he had a wife, but in my own mind I said a woman is not a He. "Who is the He?" "Have you your father or a brother?" "No! It is my wife." "Your wife is a He?" "Yes!["] There were three or four more with me when he made that stupefying answer to me. They all burst out laughing, for they knew the meaning of expression He for she, and I did not. It was the first time I heard it. It at once reminded me of the French saying, speaking of a man's wife who rules her husband, “she wears the pants and he the petticoats,” though she is not styled He but she often fulfils his functions in the line of business, but not in the line of the science by excellence, progeny !
Pants versus petticoats again appears in Handford Lennox Gordon, Laconics (1910):
When the wife wears the pants who wears the petticoats?
And again in Florence Guy Woolston, "Albertism," in The New Republic (December 14, 1921) :
One day, when I was quite a little girl, I heard my Grandmother say to a neighbor, "You can say what you like about Mr. Brown, but Mrs. Brown wears the pants."
That was strange, I thought. I looked across the street, expecting to see Mr. Brown suddenly attired in petticoats.
The earliest instance of "wears the skirt" (in the relevant sense) that a Google Books search finds is from The China Critic, volume 7 (1934) [snippet]:
She never wants to be above the man in the family or in social life. She is always content to be his wife, with all the ramifications that the word implies in a dictionary. A man never wants to let on to others that he wears the skirt in the family, though he may be doing it and liking it, and this the woman plays to the utmost.
"Wears the Pants" versus "Wears the Trousers" in British and U.S. English
Tristan R points out in a comment beneath the poster's question, "wears the trousers" is more common than "wears the pants" in British English. An Ngram Viewer chart of published British works in the Google Books database for the period 1850–2000 seems to confirm his observation, though the difference is not huge:
The corresponding Ngram Viewer chart of usage in U.S. publications shows a clear preference for "wears the pants":