Let $A \in \mathbb R^{n \times n}$ be a real matrix. I want to prove that there exists a vector $v \in \mathbb R_{\geq 0}^n$, $v \neq 0$ such that $Av \geq A^Tv$ in every coordinate.
This comes for free from an "unrelated" claim, and I would like to know what is a more natural approach to attack this problem from the first place.
(I think it might be better not to say what is this claim, for now, so as not to fixate similar directions to possible answers).
What might be an algebraic approach to this question? (or is it not "algebraic in nature", and if so why?)
As noted in a comment, your conjecture is equivalent to the statement that if $K$ is real skew-symmetric, then there is a nonzero and nonnegative vector $v$ such that $Kv\ge0$. This is a consequence of a result known as "Tucker existence lemma". See my other answer on this site.