= Solution
The defining condition for an <antisymmetric matrix> forces
$$
A=\begin{pmatrix}0&a\\-a&0\end{pmatrix},\qquad
A(x,y)^T=(ay,-ax)^T.
$$
Its <dot product> with $(x,y)^T$ is zero, and its <Euclidean norm> is $|a|\sqrt{x^2+y^2}$. Thus it combines a right-angle <rotation> with a uniform <dilation> by $|a|$. More precisely, the <rotation> is clockwise for $a>0$ and anticlockwise for $a<0$. A direction-independent description is a clockwise right-angle <rotation> followed by multiplication by the signed scalar $a$.
A second application gives
$$
\boxed{A^2=-a^2I,\qquad A^2\mathbf v=-a^2\mathbf v}.
$$
Geometrically this is a half-turn combined with a <dilation> by $a^2$. Because $A$ is nonzero, $a\ne0$.
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