let
$G=\left \{T\in GL(n, \Bbb{R}) |T^t ST=S, \text {for all skew symmetric} \space S \in M(n, \Bbb{R})\right \}$
then show that $G$ is a subgroup of $GL(n, \Bbb{R})$.
$\textbf{Try}:$
Clearly $I_n \in G$. Let $T_1,T_2\in G$ so we have to prove that $T_1 T_2^{-1} \in G$. So, for any skew symmetric matrix $S$, we have
$(T_1 T_2^{-1})^tS(T_1T_2^{-1}) =({T_2}^{-1})^t ({T_1}^tST_1){T_2}^{-1}=({T_2}^{-1})^tS{T_2}^{-1}$.
Now we have to show that,
$({T_2}^{-1})^tS{T_2}^{-1}=S$. I'm stucked here. I'm unable to justify this logically step by step. Please describe me. Thanks in advance.