The concept of a Hadamard matrix as a binary orthogonal matrix is extended to higher dimensions. Ann-dimensional Hadamard matrix[h_{ijk \cdots n}]is defined as one in which all parallel(n - 1)-dimensional layers, in any axis-normal orientation, are uncorrelated. This is equivalent to the requirements thath_{ijk \cdots n} = \pm1and that\sum_{p} \sum_{q} \sum_{r} \cdots \sum_{y} h_{pqr \cdots yb}= m^{(n-1)} \delta_{ab}where(pqr \cdots yz)represents all permutations of(ijk \cdots n). A "proper"n-dimensional Hadamard matrix is defined as a special case of the above in which all two-dimensional layers, in all axis-normal orientations, are Hadamard matrices, as a consequence of which all intermediate-dimensional layers are also Hadamard matrices. Procedures are described for deriving three- and four-dimensional Hadamard matrices of varying propriety from two-dimensional Hadamard matrices. A formula is given for a fully propern-dimensional matrix of order two, which can be expanded by direct multiplication to yield proper(2^{t})^{n}Hadamard matrices. It is suggested that proper higher dimensional Hadamard matrices may find application in error-correcting cedes, where their hierarchy of orthogonalitias permit a variety of checking procedures. Other types of Hadamard matrices may be of use in security codes on the basis of their resemblance to random binary matrices.