And by popular request, here's my comment as an answer :)
Your guess is correct. Contact structures are structures associated to a one-form $\alpha$ with maximal rank. There are two cases:
for odd rank, you want $\alpha∧(d\alpha)^k$ to be nowhere vanishing for the largest possible $k$ allowed by dimension, and
for even rank, you want the same condition on $(d\alpha)^k$.
In the former case you have a contact structure and in the latter an exact symplectic structure.
More generally, symplectic forms are nondegenerate by definition. You can understand nondegeneracy of a 2-form $\omega$ pointwise, where it turns into the statement that an antisymmetric matrix has nonzero determinant. This can only happen if the dimension is even.
I'm assuming finite-dimensionality throughout. There is a reasonably well-developed theory of infinite-dimensional symplectic manifolds and presumably also of contact manifolds.