Intersections of the space of skew-symmetric maps with its translates

For a quadratic space V over a field K, let 5 C End(V) be the space of all maps which are skew-symmetric with respect to the inner product. For gGGL(V), let (g) = dim(i? fl g). In this paper we determine the largest few values possible for 3)(g), and we classify the maps g which achieve these values. The restriction of this result to maps g in the orthogonal group 6(V) generalizes the characterization of symmetries originally proved by Botta and Pierce.