The Symplectization paper says: This cannot be so in more than 2 dimensions. Choose three independent vectors u, v and w. Consider the function
f(θ) = Ω(u, (cos θ)v + (sin θ)w).
f(0) = Ω(u, v) while f(π) = Ω(u, −v)
but Ω(u, v) = − Ω(u, −v)
and so f(0) = −f(π)
but f is real and continuous and therefore there must be φ for which f(φ)=0.
Since v and w are independent x = (cos φ)v + (sin φ)w = 0.
We have then Ω(u, x) = 0 with u not parallel to x.

I think the plainest definition of non-degenerate ω, is that for any non-zero vector u, there is a vector v such that ω(u, v) ≠ 0.