For an endomorphism $alpha$ of a ring R, we introduce the weak $alpha$-skew Armendariz rings which are a generalization of the $alpha$-skew Armendariz rings and the weak Armendariz rings, and investigate their properties. Moreover, we prove that a ring R is weak $alpha$-skew Armendariz if and only if for any n, the $n;{ imes};n$ upper triangular matrix ring $T_n(R)$ is weak $ar{alpha}$-skew Armendariz, where $ar{alpha};:;T_n(R);{ ightarrow};T_n(R)$ is an extension of $alpha$ If R is reversible and $alpha$ satisfies the condition that ab = 0 implies $a{alpha}(b)=0$ for any a, b $in$ R, then the ring R[x]/($x^n$) is weak $ar{alpha}$-skew Armendariz, where ($x^n$) is an ideal generated by $x^n$, n is a positive integer and $ar{alpha};:;R[x]/(x^n);{ ightarrow};R[x]/(x^n)$ is an extension of $alpha$. If $alpha$ also satisfies the condition that ${alpha}^t;=;1$ for some positive integer t, the ring R[x] (resp, R[x; $alpha$) is weak $ar{alpha}$-skew (resp, weak) Armendariz, where $ar{alpha};:;R[x];{ ightarrow};R[x]$ is an extension of $alpha$.