Non-negative determinant of a block matrix

Hint. Start with $$\det{\begin{bmatrix}A & B \\-B & A\end{bmatrix}}=\det{\begin{bmatrix}A-iB& B+iA \\-B & A\end{bmatrix}}.$$

Let $\left( \begin{matrix} x \\ y \end{matrix} \right) \in \mathbb{R}^{2n}$ be an eigenvector with eigenvalue $\lambda$.
Then $\left( \begin{matrix} -y \\x \end{matrix} \right)$ is also an eigenvector with eigenvalue $\lambda$, so all eigenvalues have an even number of linear independent eigenvectors.