Dinh et al. (Discrete Mathematics 341 (2018) 324-335) obtained all self-dual constacyclic codes of length $p^s$ over $R_2=\mathbb{F}_{p^{m}}+u\mathbb{F}_{p^{m}} $, where $ p $ is a prime number and $ u^2=0. $ In this paper, we determine the structure of (Euclidean) dual of some special skew cyclic codes of length $p^s$ over $R_2$, and establish all of them which are self-dual. As a special case, we conclude all of results appeared in the above paper for cyclic codes.
