We theoretically study the RKKY interaction between localized spins embedded in a 1D- or 2DEG with Rashba spin-orbit coupling. We show that rotation of spin of conduction electrons due to the Rashba spin-orbit coupling causes a twisted RKKY interaction between localized spins which consist of three different terms: Heisenberg, Dzyaloshinsky-Moriya, and Ising interactions. We also study the spin-configuration of linear- and ring-shaped artificial molecules consisting of localized spins coupled via the twisted RKKY interaction. We find that the square-norm of the total spin of an artificial molecule oscillates with the twist angle θ and has maxima at θ = an odd integer times π. The square-norm of the total spin of the ring-shaped artificial molecules change drastically at certain values of θ where the lowest two energy levels cross each other.