Z2Z4-additive codes have been defined as a subgroup of Z2^{r} x Z4^{s} in [5] where Z2, Z4 are the rings of integers modulo 2 and 4 respectively and r and s positive integers. In this study, we define a new family of codes over the set Z2^{r}[\barξ] x Z4^{s}[ξ] where ξis the root of a monic basic primitive polynomial in Z4[x]. We give the standard form of the generator and parity-check matrices of codes over Z2^{r}[\barξ] x Z4^{s}[ξ] and also we introduce skew cyclic codes and their spanning sets over this set.