By means of singular value decomposition, I think that the general answer for a real n by n matrix should be: Required volume = vol(O(n))^2*int_ 0<=s_ n<=s_ n-1<= . . .s_ 1<=1(prod_ i < j < n (s_ i^2-s_ j^2)). $$ {\rm vol}(O(n))^2 \int\limits_{0\leq s_n \leq s_{n-1}\leq \dots s_1\leq 1}\prod_{i < j < n} (s_ i^2-s_ j^2).$$
O(n) is the n-dimensional orthogonal group
