If A is a skew-symmetric matrix,

It is given that, $A$ is a skew-symmetric matrix.

$\therefore A^{T}=-A$

Now,

$\left(A^{2}\right)^{T}$

$=(A A)^{T}$

$=A^{T} A^{T}$     [For any matrices $X, Y,(X Y)^{\top}=Y^{\top} X^{\top}$ ]

$=(-A)(-A)$                     [Using (1)]

Since $\left(A^{2}\right)^{T}=A^{2}$, so the matrix $A^{2}$ is a symmetric matrix.

If $A$ is a skew-symmetric matrix, then $A^{2}$ is a symmetric matrix