## non-degenerate Poisson bracket and even-dimensional manifold

From this reference:

http://www.amazon.com/exec/obidos/tg...glance&s=books titled From Classical to Quantum Mechanics,

I quote the following: ( $$\xi^i$$ are coordinate functions)

Let M be a manifold of dimension n. If we consider a non-degenerate Poisson bracket, i.e. such that

$$\{\xi^i,\xi^j\} \equiv \omega^i^j$$

is an inversible matrix, we may define the inverse $$\omega_i_j$$ by requiring

$$\omega_i_j \omega^j^k = \delta_i^k$$

We define a tensorial quantity

$$\omega \equiv \frac{1}{2}\;\omega_i_j \; d\xi^i \wedge d\xi^j$$

which turns out to be a non-degenerate 2-form.
This implies that the dimension of the manifold M is necessarily even.

My questions are the following:

I dont understand the two statement that I have put in red above.
What is a non-degenerate 2-form?
Why does this one above 'turns out' to be non-degenerate?
Why does that imply that M is even?
Additional comments would be welcome. Like concerning the meaning of $$\omega$$ above.

In addition, I guess the point here by a shorter way: I think that all odd-dimensional antisymmetric matrices are singular. Is there a link with the language used above?