= Symplectic vector space
{wiki}
A <symplectic vector space> is a <vector space> equipped with a <nondegenerate bilinear form> $\omega$ that is an <alternating bilinear form>: $\omega(v,v)=0$ for every <vector> $v$. This implies $\omega(v,w)=-\omega(w,v)$. When the <characteristic> is two, that latter equation alone does not imply the alternating condition. A <symplectic vector space> of finite <dimension> has even <dimension>: splitting off a <hyperbolic pair> leaves a smaller <nondegenerate bilinear form>, and repeating gives a decomposition into two-dimensional planes.
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