I have the following definition for a contact structure in my lecture:
A contact structure on a manifold $W^{2n+1}$ is a hyperplane field $\xi \subset TW$ which is maximally non-integrable, meaning that the local defining forms satisfy \begin{align*} \alpha \wedge d \alpha ^n \end{align*} is a local volume form.
Now this confuses me, since a defining form means that locally, $\xi= ker \ \alpha$. Now let $v \in \xi$, then
$\alpha \wedge d \alpha ^n (v,v,.....,v) = 0$
since $\alpha(v)=0$, right? But a volume form is defined to be non-vanishing. What am I missing ?
A volume-form is non-vanishing as a smooth section of the top-exterior bundle of the cotangent bundle: $\omega \in \Omega^n(M)$ is a function $\omega: M \ni p \mapsto \omega_p \in \wedge^nT^*M$. And this function should for no $p$ evaluate to zero.
Any form, now seen as a map that evaluates vectorfields, will vanish on the zero-vectorfield. So asking for a form that doesn't vanish as a map $\omega: \Gamma(TM)^n \longrightarrow C^\infty(M)$, is non-sensical.
I think this distinction is what you were missing.