Kunze Linear Algebra · Problem 10.3.2

A skew-symmetric bilinear form is completely determined by how it behaves on pairs of basis vectors—and the constraint that $B(v,v) = 0$ for all $v$ means the diagonal entries vanish. The key insight is that any such form is really just encoding a cross product: the matrix representing $B$ must be antisymmetric, so it has only three independent entries (the upper triangle), and these three parameters correspond exactly to the coefficients of a $3 \times 3$ antisymmetric matrix. Your job is to characterize the general form and show how it relates to the structure of antisymmetric matrices.

A skew-symmetric bilinear form is completely determined by how it behaves on pairs of basis vectors—and the constraint that $B(v,v) = 0$ for all $v$ means the diagonal entries vanish. The key insight is that any such form is really just encoding a cross product: the matrix representing $B$ must be antisymmetric, so it has only three independent entries (the upper triangle), and these three parameters correspond exactly to the coefficients of a $3 \times 3$ antisymmetric matrix. Your job is to characterize the general form and show how it relates to the structure of antisymmetric matrices.