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} product_{i<j} |x_i^2-x_j^2| dx_1...dx_n

where c_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][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
64/405 pi^4 for 3x3 matrices
128/70879 pi^8 for 4x4 matrices
...

There might be a closed formula for the integral above.


  [1]: http://analysis.math.uni-kiel.de/koenig/inhalte/isotropy.pdf