Linear/additive preservers of rank 2 on spaces of alternate matrices over fields

Let Kn(F) be the linear space of all n × n alternate matrices over a field F, and let Kn2(F) be its subset consisting of all rank-2 matrices. An operator ϕ : Kn(F) → Kn(F) is said to be additive if ϕ(A + B) = ϕ(A) + ϕ(B) for any A, B ∈ Kn(F), linear if ϕ is additive and ϕ(aA) = af(A) for every a ∈ F and A ∈ Kn(F), and a preserver of rank 2 on Kn(F) if ϕ(Kn2(F))⊆Kn2(F). When n ⩾ 4, we characterize all linear (respectively, additive) preservers of rank 2 on Kn(F) over any field (respectively, any field that is not isomorphic to a proper subfield of itself).