Odd-dimensional skew-symmetric singularity theorem ($\det A=0(n\text{ odd},A^{\top }=-A)$)

An odd integer number of rows and columns forces a real skew-symmetric matrix to be a singular matrix. The determinant satisfies $\det A=\det A^{\top }=\det (-A)=(-1{)}^n\det A=-\det A$, so it is zero. The argument also works over any field of characteristic different from two.