This work focuses on the definition and study of the n-dimensional k-vector, an algorithm devised to perform orthogonal range searching in static databases with multiple dimensions. The methodology first finds the order in which to search the dimensions, and then, performs the search using a modified projection method. In order to determine the dimension order, the algorithm uses the k-vector, a range searching technique for one dimension that identifies the number of elements contained in the searching range. Then, using this information, the algorithm predicts and selects the best approach to deal with each dimension. The algorithm has a worst case complexity of $\mathcal{O}(nd(k/n)^{2/d})$, where $k$ is the number of elements retrieved, $n$ is the number of elements in the database, and $d$ is the number of dimensions of the database. This work includes a detailed description of the methodology as well as a study of the algorithm performance.