I encountered a neat problem in a problem in particle physics

So given $n$ skew symmetric matrices $A_1,...,A_n$ in $\mathbb{C}^{d \times d}.$

I would like to call this the commutator property: $A_1,...,A_n$ have the $\textbf{commutator property}$, if the only skew-symmetric matrix that has zero trace and commutes with all the $A_i$, zero trace because $i \mathbb{1}$ will of course always commute, is the zero matrix.

Now, if a matrix $B$ commutes with some $A_1,...,A_n$ then $A_i$ and $B$ are simultaneously diagonalizable, because they are normal. Unfortunately, this does not give us any good condition on all of the $A_1,...,A_n.$

So I was thinking: Assuming you have a bunch (let's say 50) of those matrices and a computer. How do you decide whether the matrices have the commutator property?