A matrix $A_N$ is called Gaussian orthogonal ensemble (GOE) if
- $A_N$ is symmetric;
- $A_N(i, i) \sim \mathcal{N}(0,2)$ for $i=1, \dots, N$ and $A_N(i, j) \sim \mathcal{N}(0,1)$ for $1 \leq i < j \leq N$;
- $(A_N(i,j))$ independent for $1 \leq i \leq j \leq N$.
It is easy to see that $A_N := \frac{1}{\sqrt{2}} (W + W^T)$ is a GOE matrix whenever $W$ is a matrix with i.i.d. $\mathcal{N}(0,1)$-entries.
However, I am wondering whether each GOE matrix has such a representation (not just in distribution)! This is not at all clear to me, because for $i < j$ we would need $$ W(i, j) = \sqrt{2} A_N(i, j) - W(j, i) $$ and whenever we choose $W(j, i) \sim \mathcal{N}(0, 1)$ independent of $A_N(i ,j)$, we would have $W(i, j) \sim \mathcal{N}(0, 2 + 1)$.