The volume of the unit ball for the spectral norm in nxn real matrices is given by the formula
c_n int_{[-1,1]^n} producti < j |x_i^2-x_j^2| dx_1...dx_n
where c_n = n! 4^{-n} product_{k=1}^n v_k^2
and v_k=pi^{k/2}/Gamma(1+k/2) is the volume of the unit ball in R^n.
A much more general formula for calculating all kind of similar quantities appears e.g. here (Lemma 1). The proof is by applying the SVD decomposition as a change of variables.
The first values are
- 2/3 pi^2 for 2x2 matrices
- 8/45 pi^4 for 3x3 matrices
- 4/1575 pi^8 for 4x4 matrices ...
There might be a closed formula for the integral above. Edit : such a formula appears in Armin's post below !!