In how many ways can $2n$ players be paired?
Your attempt would be the total number of matches needed if everyone should meet everyone.
In this case, however, you want something else. The reason the answer is what it is is the following: Player number 1 has $2n-1$ potential opponents. Whoever his opponent is, the next unassigned player has $2n-3$ to choose from. And so on. All in all, you get $(2n-1)\cdot(2n-3)\cdots3\cdot 1$.