In this paper, we study the d-dimensional rectilinear drawings of the complete d-uniform hypergraph K_2d ^d. Anshu et al. (2017) [3] used Gale transform and Ham-Sandwich theorem to prove that there exist Ω(2^d) crossing pairs of hyperedges in such a drawing of K_2d ^d. We improve this lower bound by showing that there exist Ω(2^dd) crossing pairs of hyperedges in a d-dimensional rectilinear drawing of K_2d ^d. We also prove the following results. 1. There are Ω(2^dd^3/2) crossing pairs of hyperedges in a d-dimensional rectilinear drawing of K_2d ^d when its 2d vertices are either not in convex position in R^d or form the vertices of a d-dimensional convex polytope that is t-neighborly but not (t+1)-neighborly for some constant t≥1 independent of d. 2. There are Ω(2^dd^5/2) crossing pairs of hyperedges in a d-dimensional rectilinear drawing of K_2d ^d when its 2d vertices form the vertices of a d-dimensional convex polytope that is (⌊d/2⌋−t^′)-neighborly for some constant t^′≥0 independent of d.