In formal papers, I've always been told to avoid contractions, but unlike "do not" versus "don't", I don't think that I have ever heard "of the clock" spoken aloud.
Is there a case (aside from time travel) where "of the clock" is more appropriate?
The first reference the OED has to of the clock is from Chaucer's Prologue dated 1386 (presumably they had clocks).
c1386 Chaucer Parson's Prol. 5 Ten of the clokke it was tho as I gesse.
The most recent reference to of the clock is from Gladstone speaking in Parliament in 1884
1884 Gladstonein Parlt. 26 Feb. 2/5 That the Speaker..be presented to-morrow at two of the clock in the House of Lords.
Various alternatives to of the clock have been used across the centuries. There is evidence for of clock 1473 & 1647; a clock 1584 - 1747; at the clock and simply clock 1509 - 1712. The first reference to o'clock in the OED is from Robert Southey - 1829
1829 R. Southey Roprecht 11, From noon Till ten o'clock
So, in answer to your question, there is no law against your using any expression you like, so long as it is decent and honest. But if you want to use of the clock you may be the first person to do so since the late nineteenth-century. But there is absolutely no guarantee that no one else has used it since Gladstone .
No. In Modern English, o'clock is not a contraction, and of the clock does not exist as an idiomatic expression. (It can occur literally, for example I saw it on top of the clock; but it doesn't exist in the sense of telling the time).
The phrase 'o'clock' is a linguistic fossil, and is never written as 'of the clock'.
It actually dates from the 14th century when clocks first started to be installed in churches to tell the hours. Before that, time had been computed by the position of the sun - there were twelve hours to the day and twelve to the night, so winter daytime hours were much shorter than summer daytime hours and winter nighttime hours. But hours in clock time are always the same length; so when you mentioned a time in the 14th century you needed to specify whether you meant 'three by the sun' or 'three of the clock', which could be very different times! But now we have been going by clock time for more than half a millennium, 'o'clock' no longer has that literal meaning and is just a tag to indicate that you are referring to the time.
Of the clock is used to mark time well into the 20th century, though it is largely limited to legislative record-keeping:
The phrase also appears in At Swim-Two-Birds, a 1939 novel by Irish writer Brian O'Nolan:
I was acquainted of the way by angels, said the cleric, and the ladder [was . . .] conveyed to my college in a sky-carriage in the middle of last night, at two of the clock to speak precisely. (source)
The difficulty in finding these examples, however, seems proves the opposite point: that of the clock, for the purposes of keeping time, is long obsolete, especially outside the formality of legislative bodies.
For a formal paper I would use 9:00 rather than 9 o'clock, and '9 of the clock' is never used in practice.
Building on the answers relating to the use of the expression in the UK Parliament, it's interesting to note that both the validity of the expression "… of the clock" and its archaism were brought out in this (perhaps vaguely humorous?) reference in the House of Commons in 2001:
"By 3.30 this afternoon, or half-past three of the clock as my right honourable Friend the Member for Bromley and Chislehurst (Mr. Forth) more correctly describes it, we are supposed to have considered nine detailed Government amendments."