Diameter preserving surjection on alternate matrices

Let F be a field with vertical bar F vertical bar >= 3, K-m be the set of all m x m (m >= 4) alternate matrices over F. The arithmetic distance of A, B is an element of K-m is d(A, B) := rank (A - B). if d(A, B) = 2, then A and B are said to be adjacent. The diameter of K-m is max{d(A, B) : A, B is an element of K-m}. Assume that (sic) : K-m -> K-m is a map. We prove the following are equivalent: (a) (sic) is a diameter preserving surjection in both directions, (b) (sic) is both an adjacency preserving surjection and a diameter preserving map, (c) (sic) is a bijective map which preserves the arithmetic distance.