  Let $X$ be a symmetric random matrix with independent but non-identically
distributed centered Gaussian entries. We show that $$
  \mathbf{E}\|X\|_{S_p} \asymp
  \mathbf{E}\Bigg[
  \Bigg(\sum_i\Bigg(\sum_j X_{ij}^2\Bigg)^{p/2}\Bigg)^{1/p}
  \Bigg] $$ for any $2\le p\le\infty$, where $S_p$ denotes the $p$-Schatten
class and the constants are universal. The right-hand side admits an explicit
expression in terms of the variances of the matrix entries. This settles, in
the case $p=\infty$, a conjecture of the first author, and provides a complete
characterization of the class of infinite matrices with independent Gaussian
entries that define bounded operators on $\ell_2$. Along the way, we obtain
optimal dimension-free bounds on the moments $(\mathbf{E}\|X\|_{S_p}^p)^{1/p}$
that are of independent interest. We develop further extensions to
non-symmetric matrices and to nonasymptotic moment and norm estimates for
matrices with non-Gaussian entries that arise, for example, in the study of
random graphs and in applied mathematics.
