Alternating forms and self-adjoint operators

Let k be a field and n⩾1 an integer. We study the action of the symplectic group over k on the set of alternating forms on k2n. We show that the action on pairs of forms can be interpreted in terms of the conjugation action on self-adjoint operators, and obtain some old and new results using this interpretation. In particular, for each k and n, we determine the smallest base size for the action of PGL2n(k) on the set of non-degenerate skew-symmetric matrices over the field k, modulo scalars.