Adjoint of skew-symmetric matrix

Question

If $A$ is a skew-symmetric $n\times n$ matrix, verify that $\operatorname{adj} A$ is symmetric or skew-symmetric according to whether $n$ is odd or even.

Things I can think of is $A^T=-A$ for skew-symmetric matrix, and the other is $\operatorname{adj} A=(\operatorname{cofactor}A)^T$.

Answer 1

Here are some hints. Show that $$\operatorname{adj}(A^T)=(\operatorname{adj} A)^T$$ for every square matrix $A$. Then, show that $$\operatorname{adj}(-A)=(-1)^{n-1}\operatorname{adj} A$$ for every square matrix $A$ of order $n$.