For an endomorphism α of a ring R, we introduce the weak α-skew Armendariz rings which are a generalization of the α-skew Armendariz rings and the weak Armendariz rings, and investigate their properties. Moreover, we prove that a ring R is weak α-skew Armendariz if and only if for any n, the n × n upper triangular matrix ring Tn(R) is weak ¯α-skew Armendariz, where ¯α : Tn(R)→ Tn(R) is an extension of α. If R is reversible and α satisfies the condition that ab = 0 implies aα(b)=0 for any a, b ∈ R, then the ring R[x]/(xn) is weak ¯α-skew Armendariz,where (xn) is an ideal generated by xn, n is a positive integer and ¯α : R[x]/(xn)→ R[x]/(xn) is an extension of α. If α also satisfies the condition that αt = 1 for some positive integer t, the ring R[x] (resp,R[x; α]) is weak ¯α-skew (resp, weak) Armendariz, where ¯α : R[x] → R[x]is an extension of α.