Given A $\in M_{nxn}(\mathbb{R})$ an anti symmetric matrix $(A^T = -A)$, lets write its columns as : $C_1,...,C_n$, we want to show that there exists n non-negative numbers: $a_1 ,..., a_n$ (which are not all 0) such that $a_1C_1+...+a_nC_n$ is a vector with non-negative coordinates.
I first thought to define the set P as all vectors with non-negative coordinates (over real numbers) and if we can find a vector $v \in P$ with $Av$ is in P then we are done, otherwise I wanted to look over the set $M_{nxn} (\mathbb{R}) $\P and infer the existence over there- I'm not sure how to continue or if the beginning of my solution is valid
I'll accept any solution there is, thanks in advance
Your question is very interesting. Skew symmetry is preserved under orthogonal similarity, but being entrywise nonnegative is a basis dependent property. I have yet to understand how to reconcile these two seemingly conflicting properties.
Anyway, you are essentially asking that when $A$ is real skew symmetric, whether there exists a nonnegative and nonzero vector $x$ such that $Ax\ge0$. I'm not sure if there is a simpler and more direct proof, but a stronger result by Tucker (1956) states that there actually exists some $x\ge0$ such that $Ax\ge0$ and $Ax+x>0$.
See, e.g. Tucker's theorem from Farkas lemma on this site, or lemma 3 (Tucker existence lemma) on p.15 of Giorgio Giorgi (2014), Again on the Farkas Theorem and the Tucker Key Theorem Proved Easily.