During my researches I've come across the following question.

Let $A$ and $B$ be a couple of square $k\times k$ skew symmetric matrices on $\mathbb R$. Let us consider the (real) pencil generated by $A$ and $B$ that is $$P(A,B)\doteq \lambda A+\eta B,\; \lambda,\eta\in\mathbb R.$$

What I am studying is the stabilization of $P(A,B)$ under the action of the orthogonal group $O(k)$, that is I would like to know what can be deduced about $A$ and $B$ under the following request that is

$$(*)\quad M P(A,B) M^{T}\subset P(A,B),\quad \forall M\in O(k).$$

Another interesting question for me would be to characterize all the $M\in O(k)$ such that $(*)$ holds.

What I would like to know is if there are already similar results in literature and what is known on the subject, references are warmly welcomed and I thank you all in advance for your kind help.

Best Wishes.