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Let $\omega$ be a nondegenerate alternating $2$-form on an $2n$-dimesional Vectorspace $V$, meaning that for all nonzero $v \in V$ the map $w \mapsto \omega(v,w)$ is not identically zero.

Why is the nth power $\omega^n = \omega \wedge \dots \wedge \omega \neq 0$?

Every textbook i have looked at says this is very easy, but i do not see it.

Thanks for your help.

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    The answer of Jason DeVito shows that $\omega$ need not be closed for this to hold. The nondegeneracy condition just is equivalent to the condition that $V$ has an even dimension and $\omega^n$ is nowhere vanishing on $V$. So this answeres your question completely. – Dietrich Burde Jun 12 '15 at 12:18

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