I want to describe that school life is full of continuous competition, should I say "School life is continuous competition" or "Scool life is a continuous competition" ?
You could use either. The two meanings are subtly different but probably amount to the same thing in this instance.
I'll refer to this dictionary page: http://www.oxforddictionaries.com/definition/english/competition
1) School life is continuous competition
This is using defn 1: "The activity or condition of striving to gain or win something by defeating or establishing superiority over others:"
Ie, it's like saying "While we are at school we are continuously competing with others".
2) School life is a continuous competition
This is using defn 1.1: "an event or contest in which people take part in order to establish superiority or supremacy in a particular area:"
It's like saying "Being at school is like being in a competition".
So, the second one is a simile (you're not actually in a competition, even if it feels like it), while the first one is a more direct description. Of course, school could be described as literally being a competition between students, especially if there are prizes, eg for best student in the year or whatever. This makes the second one more literally true and less of a simile.
However, I think that most people would take the meaning to be the same in either case.