Show that $A\in M_n(\mathbb R)$ is skew-symmetric if $\det(A+M)=0$ for all M skew-symmetric, with $n$ odd.

Question

Let $n$ be an odd integer, and $A\in M_n(\mathbb R)$ such that, for all $M\in M_n(\mathbb R)$ skew-symmetric, we have: $$ \det(A+M)=0$$ Show that $A$ is skew-symmetric.

What I did so far: Nothing much really…

• $\det(A) = 0$ plugging $M=0$
• $\det\left(\dfrac{A+A^T}{2}\right) = 0$ plugging $M=\dfrac{A^T-A}{2}$
• $\det(M)=0$ for all $M$ skew-symmetric, since $n$ is odd