Let (V, b) be a finite-dimensional vector space over an arbitrary field k together with a nondegenerate symmetric or skew-symmetric bilinear form.

Non-degenerate skew-symmetric bilinear forms can only exist on even-dimensional vector spaces, so symplectic manifolds necessarily have even dimension.

This defines a symmetric bilinear form in case and a skew-symmetric bilinear form in case .

In linear algebra, skew-Hamiltonian matrices are special matrices which correspond to skew-symmetric bilinear forms on a symplectic vector space.

An almost symplectic manifold is a differentiable manifold equipped with a smoothly varying non-degenerate skew-symmetric bilinear form on each tangent space, i.e., a nondegenerate 2-form ω, called the symplectic form.