On the vector space of matrices equipped with the p-Schatten norm, consider the unit ball normalized to have Lebesque volume 1. Let $ W$ be the random matrix uniformly distributed on this set. We compute sharp upper and lower bounds for the moments of marginals of the random matrix $W$. As an application, we characterize subgaussian and supergaussian directions, estimate the volume of sections of these balls, and provide precise tail estimates for the singular values of the matrix $W$.  Based on a joint work with Kavita Ramanan.