Let R be a prime ring of characteristic different from 2 and 3,Qr be its right Martindale quotient ring and C be its extended centroid.Suppose that F and G are generalized skew derivations of R,L a non-central Lie ideal of R and n ≥ 1 a fixed positive integer.Under appropriate conditions we prove that if (F(x)x-xG(x))n =0 for all x ∈ L,then one of the following holds:(a) there exists c ∈ Qr such that F(x) =xc and G(x) =cx;(b) R satisfies s4 and there exist a,b,c ∈ Qr such that F(x) =ax + xc,G(x) =cx + xb and (a-b)2 =0.