I had the computer do it for me :)

Really though, the way to do this is to represent each game as a payout matrix A, which for this category of game will be skew-symmetric with -1/0/+1 entries. Then you find the null space of that matrix, impose the constraint that probabilities must sum to 1 and individually be >= 0, and that gives you the endpoints of the Nash Equilibria. The optimal strategies (could be one unique one for odd n!) lie on the convex hull of those points.