Gaucheness is the noun: the awkward, socially clumsy quality itself. You can talk about someone’s gaucheness at a formal dinner — the way they keep choosing the wrong fork, or saying too much, not out of unkindness but because the room’s quiet rules are still new. It is the same idea as gauche, only now you are naming the trait rather than describing the person. It is a bookish word. In ordinary speech people are more likely to say awkwardness, or just “that was a bit awkward.” Though actually, gaucheness still points at manners and social rules, not at being bad at sport or at schoolwork.

不得體、舉止上的笨拙