Let $V$ be a complex vector space, and let $\mathbb{P}(V)$ denote the projectivization of $V$ (i.e. space of 1-dimensional subspaces, i.e. 1st Grassmanian). Suppose further that $V$ is endowed with a non-degenerate two-form $\omega$.
We know that the tangent space $T_v\mathbb{P}(V)$ is canonically isomorphic to $V/\left<v\right>$. We can thus define a 1-form $\alpha$ by the maps
$$\alpha_v(\cdot)=\omega(v,\cdot),$$
which is well defined since $\omega(v,v)=0$. My question is how I can explicitly compute the exterior derivative $d\alpha$. The reason I ask this is that I am trying to do Exercise 10.1 in Voisin's "Hodge theory and complex algebraic geometry i", where they ask to show that the top dimensional form $\alpha \wedge (d\alpha)^{n-1}$ does not vanish at any point, and I feel that I can't do the exercise if I don't know how to compute $d\alpha$.
EDIT: Below is the exercise in question.
Let $V$ be a complex vector space endowed with a non-degenerate 2-form $\omega$. Recall that $T_{\mathbb{P}(V),v}$ is isomorphic to $V/\left<v\right>$. Show that the form $\alpha$ defined (up to multiplicative coefficient) by $$\alpha_v(\cdot)=\omega(v,\cdot)$$ provides a contact structure on $\mathbb{P}(V)$
Contact structures are defined above as follows:
Let $X$ be a complex manifold of dimension $2n-1$ a contact structure on $X$ is determined by the local datum of a holomorphic $1$-form $\alpha$ which is well defined up to multiplication by an invertible holomorphic function and which satisfies the condition that the $(2n-1)$-form $$\alpha\wedge (d\alpha)^{n-1}\in K_X$$ does not vanish at any point.