Matrix-vector multiplication/cross product problem

OK, let's put it other way as $mathbf{w}times mathbf{v}=-mathbf{A}mathbf{w}$. We can write the the cross product as vector-matrix multiplication: $$mathbf{w}timesmathbf{v} =[mathbf{w}]_times mathbf{v}=begin{bmatrix},0&!-w_{3}&,,w_{2}\,,w_{3}&0&!-w_{1}\-w_{2}&,,w_{1}&,0end{bmatrix}mathbf{v}.$$ So you can write your equation as a system of linear equations $$[mathbf{w}]_times mathbf{v}=-mathbf{A}mathbf{w}.$$ Matrix $[mathbf{w}]_times$ has rank $2$ and its nullspace is spanned by $[w_1,,w_2,,w_3]^top$.

Now depending on whether you assume $w_2neq 0$ or $w_3neq 0$, you can transform this system and find a particular solution. However, this solution can be found only if $langlemathbf{w},mathbf{Aw}rangle=0$. In particular, this implies that $mathbf{A}^top=-mathbf{A}$.