Assume we have $A, B \in \mathbb{S}_n(\mathbf{R})$, where $\mathbb{S}_n(\mathbf{R})$ is the set of all symmetric matrices with real entries and size $n$. The Lie-bracket (or, the commutator for a matrix group) is defined as follows $$ [A, B] = AB - BA. $$ Now assume $A, B$ are random matrices (feel free to assume any particular ensemble of matrices, say Gaussian ensembles). Are there such "special" random matrices that something could be said about $[A, B]$? Maybe some bounds on its operator norm?
Any other properties along with suggestions and comments will be highly appreciated.
EDIT (Aug 4, 2018): Notice that $(AB)_{ij} = (BA)_{ji}$ yielding the following simple fact: $[A, B]_{ij} = -[A, B]_{ji}$. The latter, in particular, implies that $[A, B]_{ii} = 0$ as noticed by daruma in comments.
So, the lie bracket $[A, B]$ can be rewritten as follows: $$ [A, B] = \begin{pmatrix} 0 & & S \\ & \ddots\\ -S^T & & 0\end{pmatrix} := S_1 - S_2, $$ where $S$ a diagonal block and matrices $S_1$ and $S_2$ can be viewed as upper and lower triangular matrices with corresponding blocks of $S$ and $S^T$, respectively.