Let R be a prime ring of characteristic different from 2 and 3, Q r its right Martindale quotient ring, C its extended centroid, L a non-central Lie ideal of R and n ≥ 1 a fixed positive integer. Let α be an automorphism of the ring R . An additive map D : R → R is called an α -derivation (or a skew derivation) on R if D ( xy ) = D ( x ) y + α ( x ) D ( y ) for all x , y ∈ R . An additive mapping F : R → R is called a generalized α -derivation (or a generalized skew derivation) on R if there exists a skew derivation D on R such that F ( xy ) = F ( x ) y + α ( x ) D ( y ) for all x , y ∈ R . We prove that, if F is a nonzero generalized skew derivation of R such that F ( x )× F ( x ), x n = 0 for any x ∈ L , then either there exists λ ∈ C such that F ( x ) = λ x for all x ∈ R , or R ⊆ M 2 ( C ) and there exist a ∈ Q r and λ ∈ C such that F ( x ) = ax + xa + λ x for any x ∈ R .