Assume that $\mathbf{x} \sim \mathcal{N}(0,\mathbf{I}_n)$ and $\mathbf{y}\sim \mathcal{N}(0,\mathbf{I}_p)$ are two independent standard Gaussian vectors.
What is the distribution of their outer product $$\mathbf{x}\mathbf{y}^T=(x_iy_j)_{i\leq n, j\leq p},$$ which is a $n\times p$ matrix?
In the simple case where $p=n=1$, we end up we the normal-product distribution, but in higher dimensions, things appear to get trickier and I don't know much about matrix variate distributions.
If $n=p$, then since $2 xy^\top + 2 yx^\top = (x+y)(x+y)^\top - (x-y)(x-y)^\top$ (polarization identity), and $x+y$ is decorrelated (and thus independent) from $x-y$, at least you can say that the symmetrization of $x y^\top$ (that is $xy^\top + yx^\top$) is distributed as the difference of two independent Wishard distribution (which is not a Wishard distribution).