Nothing wrong with a woman being the one that approaches the man, it doesn't come across as desperate at all. In fact, personally I find it very attractive; but that's because I find confidence, assertiveness attractive.
As a man that lacks confidence in asking women out, it's meant that possibly some girls I've liked may have liked me too but both of us ended up waiting for the other to make the first move, but that never happened.
So take the opportunity, it's better than regret.