Let R be a noncommutative prime ring of characteristic not 2 with extended centroid C, the maximal right ring of quotients Q and a nonzero generalized skew derivation $$\delta $$ δ . Assume that $$f(X_{1},\ldots ,X_{n})$$ f(X1,…,Xn) is a multilinear polynomial over C that is not central-valued on R and f(R) is the set of all evaluations of the multilinear polynomial $$f\big (X_{1},\ldots ,X_{n}\big )$$ f(X1,…,Xn) in R. Denote the set $$S:= \big \{\delta (u)u \mid u \in f(R) \big \}$$ S:={δ(u)u∣u∈f(R)} . The goal of the paper is to study $$C_R(S)$$ CR(S) , the centralizer of S in R. To be precise, given a noncentral element $$b\in R$$ b∈R it is proved that if $$b\in C_R(S)$$ b∈CR(S) , i.e., $$\begin{aligned}{}[\delta (u)u, b]=0 \end{aligned}$$ [δ(u)u,b]=0 for all $$u \in f(R)$$ u∈f(R) , then there exists $$a \in Q$$ a∈Q with $$[a, b]=0$$ [a,b]=0 such that $$\delta (x)=ax$$ δ(x)=ax for all $$x\in R$$ x∈R and $$f\big (X_{1},\ldots ,X_{n}\big )^{2}$$ f(X1,…,Xn)2 is central-valued on R. As applications to the theorem, we consider the case of $$\delta (u)u \in C$$ δ(u)u∈C for all $$u \in f(R)$$ u∈f(R) and also we investigate commuting values of two generalized skew derivations having different associated skew derivations on multilinear polynomials.