The sum of two skew-symmetric matrices is always matrix.

Let $A$ and $B$ be two skew-symmetric matrices.

$\therefore A^{\top}=-A$ and $B^{\top}=-B$     ......(1)

Now,

$(A+B)^{T}$

$=A^{T}+B^{T}$

$=-A-B \quad[$ From (1) $]$

$\therefore(A+B)^{T}=-(A+B)$

Thus, the sum of two skew-symmetric matrices is always skew-symmetric matrix.

The sum of two skew-symmetric matrices is always skew-symmetric matrix.