The difference of two different skew-symmetric matrices is:

A matrix is skew-symmetric if $A^T = -A$ (the transpose equals the negative of the original matrix).

For example, if $A = \begin{bmatrix} 0 & 2 \\ -2 & 0 \end{bmatrix}$, then $A^T = \begin{bmatrix} 0 & -2 \\ 2 & 0 \end{bmatrix} = -A$

Diagonal elements are always zero, and elements flip signs across the diagonal.

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Let $A$ and $B$ be two skew-symmetric matrices where $A^T = -A$ and $B^T = -B$.

Let $C = A - B$

Taking the transpose:

$C^T = (A - B)^T$

$C^T = A^T - B^T$

$C^T = (-A) - (-B)$

$C^T = -A + B$

$C^T = -(A - B)$

$C^T = -C$

This shows $C$ is skew-symmetric.

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The difference of two different skew-symmetric matrices is a skew-symmetric matrix.