A symplectic form on a Smooth Manifold is a smooth closed 2-Form on which isnondegenerate such that at every point , the alternating bilinear form on the Tangent Space isnondegenerate.
A symplectic form on a Vector Space over is a function (defined for all and taking values in ) which satisfies
and
Symplectic forms can exist on (or ) only if (or ) is Even-dimensional.