As far as I know, contact structures are studied from a "cohomology/dual" perspective, meaning mostly from the perspective of contact forms and their respective kernels, instead of from a "homological" version using either contact vector fields or Reeb vector fields. Is there a good reason for this? Thanks.
I agree with the previous answer that the non-degeneracy condition is very nicely expressable in terms of an annihilating form. The "homological" point of view is used however, in particular, if one is mainly interested in the "contact distribution" (i.e. the subbundle in the tangent bundle annihlated by a contact form.) This can be implemented either by allowing replacing the contact form by an equivalence class of contact forms up to multiplication by a nowhere vanishing function, or you can discuss it directly from the point of view of distributions. (The Reeb field is not to useful, since it is not possible to reconstruct the distribution from a Reeb fielld without knowing the contact form. Indeed, equivalent contact forms lead to different Reeb fields.)
From the distribution point of view, you would take a hyperplane subbundle $H$ in $TM$, and consider the quotient bundle $Q:=TM/H$ with projection $q:TM\to Q$. Then for vector fields $\xi$ and $\eta$ on $M$ which have values in $H$, you consider the projection $q([\xi,\eta])$ of their Lie bracket. One immediately verifies that the resulting map is bilinear of smooth functions, so there is an induced bundle map $\mathcal L:H\times H\to Q$. The condition that $H$ is a contact distribution then is that $\mathcal L$ is non-degenerate (as a bilinear form) in each point. This can also be phrased as the fact that $\mathcal L$ in each point $x$ makes $H_x\oplus Q_x$ into a Heisenberg-algebra.
The whole story admits a generalization to arbitrary distributions (and in general, the "homological" point of view certainly is not more complicated than the "cohomological" one). There you obtain in each point a nilpotent graded Lie algebra called the symbol algebra of the distribution in that point.