The volume of the unit ball for the spectral norm in nxn real matrices is given by the formula

$$
c_n \int\limits_{[-1,1]^n} \prod_{i < j} |x_i^2-x_j^2| dx_1\dots dx_n
$$

where $c_n = n! 4^{-n} \prod_{k=1}^n v_k^2$

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] (Lemma 1). The proof is by applying the SVD decomposition as a change of variables.

The first values are

 - 2/3 &pi;<sup>2</sup> for 2x2 matrices
 - 8/45 &pi;<sup>4</sup> for 3x3 matrices
 - 4/1575 &pi;<sup>8</sup> for 4x4 matrices
...

There might be a closed formula for the integral above.
Edit : such a formula now appears in [Armin's answer](https://mathoverflow.net/a/3151/14094)!!


  [1]: https://web.archive.org/web/20161006215040/https://analysis.math.uni-kiel.de/koenig/inhalte/isotropy.pdf