Suppose $A$ is a $n \times n$ random matrix with centered Gaussian (real) i.i.d. entries with variance $\frac{\sigma^2}{n}$.

What to we know about the spectral norm $s(A)$ of $A$, that is $\sqrt{\rho(A^t A)}$? Here, $\rho(\cdot)$ denotes the largest eigenvalue of a matrix.

In particular, if $A$ is symmetric, we know that $s(A)$ is precisely equal to $\rho(A)$ and from the circular law it implies that $s(A)$ converges to $\sigma$ as $n \to \infty$.

Is this last statement true for non-symmetric $A$?