Answer
See below
Work Step by Step
Given $A$ is an $n \times n$ skew-symmetric matrix and $n$ is odd
We have $A^T=-A \\ \rightarrow \det(A^T)=\det(-A)$
On the other hand, we also have $\det(A^T)=\det(A)\\
\det(-A)=(-1)^n\det(A)\\
\rightarrow \det(A)=-\det(A)\\
\rightarrow \det(A)=0$
