A skew-morphism of a finite group $G$ is a permutation $σ$ on $G$ fixing the identity element, and for which there exists an integer function $π$ on $G$ such that $σ(xy)=σ(x)σ^{π(x)}(y)$ for all $x,y\in G$. It has been known that given a skew-morphism $σ$ of $G$, the product of $\langle σ\rangle$ with the left regular representation of $G$ forms a permutation group on $G$, called the skew-product group of $σ$. In this paper, the skew-product groups of skew-morphisms of finite elementary abelian $p$-groups are investigated. Some properties, characterizations and constructions about that are obtained.