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http://mathworld.wolfram.com/SymplecticForm.html,

Complex Hilbert space, with "the inner-product" I<x,y> , where <.,.> is the inner-product

Does this refer to taking the imaginary part of the standard inner-product ? If so, is

I<x,y> symplectic in Complex Hilbert Space? It is obviously bilinear, but I don't see

how it is antisymmetric , i.e., I don't see that I<x,y>=-I<y,x>

Am I missing something?

Thanks.