  Denote $\Sigma_n$ and $Q_n$ the set of all $n \times n$ symmetric and
skew-symmetric matrices over a field $\mathbb{F}$, respectively, where
$char(\mathbb{F})\neq 2$ and $\lvert \mathbb{F} \rvert \geq n^2+1$. A
characterization of $\phi,\psi:\Sigma_n \rightarrow \Sigma_n$, for which at
least one of them is surjective, satisfying
$$\det(\phi(x)+\psi(y))=\det(x+y)\qquad(x,y\in \Sigma_n)$$ is given.
Furthermore, if $n$ is even and $\phi,\psi:Q_n \rightarrow Q_n$, for which
$\psi$ is surjective and $\psi(0)=0$, satisfy
$$\det(\phi(x)+\psi(y))=\det(x+y)\qquad(x,y\in Q_n),$$ then $\phi=\psi$ and
$\psi$ must be a bijective linear map preserving the determinant.
