alternating (skew) bilinear form

An alternating bilinear form is the opposite extreme from symmetric: every vector is 'perpendicular to itself' in the form's eyes, B(v, v) = 0 for all v. Expand B(u+w, u+w) = 0 using bilinearity and you instantly get B(u, w) = -B(w, u), the skew-symmetry that gives the form its other name. So alternating means the matrix is skew, A^T = -A, with zeros forced down the diagonal.

Geometrically these forms measure signed area, not length. The model is the 2x2 determinant: B(u, v) = u_1 v_2 - u_2 v_1 is the oriented area of the parallelogram spanned by u and v, and it flips sign when you swap u and v exactly because area is oriented. There is no notion of 'a vector's length' here, which is why you cannot build a quadratic form from an alternating form — Q(v) = B(v,v) is identically zero.

Their classification is strikingly clean. Any alternating form can be brought by congruence to a block-diagonal matrix made of standard hyperbolic 2x2 blocks [0, 1; -1, 0] followed by a block of zeros. The number of hyperbolic blocks is the rank divided by two — so the rank of an alternating form is always even, and rank alone is a complete congruence invariant. There is no signature to worry about.

Caveat on words: over a field of characteristic 2, 'alternating' (B(v,v)=0) is strictly stronger than 'skew-symmetric' (A^T = -A), because there -1 = 1. Outside characteristic 2 the two notions coincide, and most courses use the terms interchangeably. The nondegenerate case of these forms is so important it gets its own name: symplectic.