If $A$ and $B$ are skew-symmetric, then $A^2BA$ is symmetric

Question

Let $A$ and $B$ be skew-symmetric $3 \times 3$ matrices. That is, $A^t = -A$ and $B^t = -B$.

According to WolframAlpha, $A^2BA$ is a symmetric matrix. I believe there is an easier way to prove this fact than directly computing the matrix $A^2BA$. I tried
$$ (A^2BA)^t = A^t B^t (A^t)^2 = ABA^2, $$
but it does not help. Could you help me? Thank you.

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(Edit) Thanks to @BrunoB, I corrected my input.

Using WolframAlpha, we have the following.

Answer 1

$A(x)=a\times x$ and $B(x)=b\times x$ for some vectors $a,b.$

$ABA(x)=-(b\cdot a)(a\times x).$

Whence $ABA$ commutes with $A.$