If $A \in M_n(\mathbb C)$ is skew symmetric, $adj(A)$ is symmetric or skew symmetric depending on $n$ being even or odd

Question

A symmetric matrix is a square matrix that is equal to its transpose.

A skew-symmetric (or antisymmetric) matrix is a square matrix whose transpose is also its negative.

1. Assume that $A \in M_n(\mathbb C)$ is a skew symmetric matrix. Prove that $adj(A)$ is symmetric or skew symmetric depending on $n$ being even or odd.
2. If $A \in M_n(\mathbb C)$ is skew symmetric and $n$ is odd, then $A$ is not invertible.

Note : I wrote $(adjA)^T=adj(A^T)=adj(-A)$ but i don't know what happens next.