  We consider pairs of anti-commuting $2p$-by-$2p$ Hermitian matrices that are
chosen randomly with respect to a Gaussian measure. Generically such a pair
decomposes into the direct sum of $2$-by-$2$ blocks on which the first matrix
has eigenvalues $\pm x_j$ and the second has eigenvalues $\pm y_j$. We call $\{
(x_j, y_j) \}$ the skew spectrum of the pair. We derive a formula for the
probability density of the skew spectrum, and show that the elements are
repelling.
