We propose a new representation of k-partite, k-uniform hypergraphs, that is, a hypergraph with a partition of vertices into k parts such that each hyperedge contains exactly one vertex of each type; we call them k-hypergraphs for short. Given positive integers ℓ,d, and k with ℓ≤d−1 and k={d\choose\ell}, any finite set P of points in \mathbb{R}^d represents a k-hypergraph G_P as follows. Each point in P is covered by k many axis-aligned affine \ell-dimensional subspaces of \mathbb{R}^d, which we call \ell-subspaces for brevity and which form the vertex set of G_P. We interpret each point in P as a hyperedge of G_P that contains each of the covering \ell-subspaces as a vertex. The class of \emph{(d,\ell)-hypergraphs} is the class of k-hypergraphs that can be represented in this way. The resulting classes of hypergraphs are fairly rich: Every k-hypergraph is a (k,k-1)-hypergraph. On the other hand, (d,\ell)-hypergraphs form a proper subclass of the class of all k-hypergraphs for \ell<d-1. In this paper we give a natural structural characterization of (d,\ell)-hypergraphs based on vertex cuts. This characterization leads to a poly\-nomial-time recognition algorithm that decides for a given k-hypergraph whether or not it is a (d,\ell)-hypergraph and that computes a representation if existing. We assume that the dimension d is constant and that the partitioning of the vertex set is prescribed.