I hope this is not too elementary.
Let $B: V\ \times V \to W$ be a skew-symmetric bilinear map where $V$, $W$ are finite dimensional real vector spaces. Assume that $B (u, v)$ is never zero for linearly independent $u$, $v\in V$.
What can we say about the dimensions of $V$ and $W$?
An elementary estimate is
$$ \dim W \geq \dim V - 1 $$
In fact, for $u\in V$, $u\neq0$, the linear map $B_u : V\to W$, $B_u(v)=B(u,v)$, is injective on a codimension $1$ subspace of $V$. Can the above estimate be improved? Recall that the image of a bilinear map does not need to be a subspace.