Congruence of matrix spaces, matrix tuples, and multilinear maps

Two matrix vector spaces $V,Wsubset mathbb C^{ntimes n}$ are said to be equivalent if $SVR=W$ for some nonsingular $S$ and $R$. These spaces are congruent if $R=S^T$. We prove that if all matrices in $V$ and $W$ are symmetric, or all matrices in $V$ and $W$ are skew-symmetric, then $V$ and $W$ are congruent if and only if they are equivalent. Let $F: Utimesdotstimes Uto V$ and $G: Utimesdotstimes Uto V$ be symmetric or skew-symmetric $k$-linear maps over $mathbb C$. If there exists a set of linear bijections $varphi_1,dots,varphi_k:Uto U$ and $psi:Vto V$ that transforms $F$ to $G$, then there exists such a set with $varphi_1=dots=varphi_k$.