Let T be a factor von Neumann algebra acting on complex Hilbert space with dim(T) ≥ 2. For any T,T1,T2, . . . ,Tn ∈ T, define q1(T) = T, q2(T1,T2) = T1 ⋄ T2 = T1T∗2 + T2T∗1 and qn(T1, . . . ,Tn) = qn−1(T1, . . . ,Tn−1) ⋄ Tn for all integers n ≥ 2. In this article, we prove that a map ζ : T → T satisfies ζ(qn(T1, . . . ,Tn)) = ∑n i=1 qn(T1, . . . ,Ti−1, ζ(Ti),Ti+1, . . . ,Tn) for all T1, . . . ,Tn ∈ T if and only if ζ is an additive ∗-derivation.