Q.Show that $A=\begin{pmatrix} 0 & 2 & -3 \\ -2 & 0 & 4 \\ 3 & -4 & 0 \end{pmatrix}$ is skew-symmetric.

A matrix is skew-symmetric when its transpose equals its negative, which also forces zero diagonal entries.

Take the transpose (rows become columns): $A^{\mathsf{T}}=\begin{pmatrix} 0 & -2 & 3 \\ 2 & 0 & -4 \\ -3 & 4 & 0 \end{pmatrix}.$

Compare with $-A$. Negating every entry of $A$: $-A=\begin{pmatrix} 0 & -2 & 3 \\ 2 & 0 & -4 \\ -3 & 4 & 0 \end{pmatrix}.$ …

Some students check only that the diagonal is zero and stop. A zero diagonal is necessary but not sufficient — you must also verify a_ij = …