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

O(n) is the n-dimensional orthogonal group