I read that the necessary condition for a manifold to be symplectic is that it should be even dimensional. The justification give is that an odd dimensional skew symmetric matrix is singular. Therefore there exists a $\omega^j$ for which $\omega_{ij}u^j = 0$. This means that for all $v$ $\epsilon$ $T_PM$,
$\omega(v\otimes u)=\omega_{ij}v^iu^j=0$
How does odd dimensional skew symmetric matrix being singular leads to the existence of such a $u^j$
