Does every real skew symmetric matrix map some non-zero vector with non-negative entries into a vector with non-negative entries?

No such matrix exists. Tucker's existence lemma asserts that if $A$ is skew-symmetric, then there exists a vector $x$ such that $x\ge0,\,Ax\ge0$ and $Ax+x>0$ (so that $x$ must be nonzero). See Giorgio Giorgi (2014), Again on the Farkas Theorem and the Tucker Key Theorem Proved Easily, p.15, lemma 3.