If we do symplectic linear algebra on a finite-dimensional vector space $V$, then what does $$\omega(v,w) \neq 0$$ or $$\omega(v,w) = 0$$ actually tell us about the vectors $v,w$? ($\omega$ is the skew-symmetric non-degenerate bilinear form) Afais, the first property tells us that they are linearly independent.
If anything is unclear, please let me know.