n-dimensional fuzzy sets are an extension of fuzzy sets where the membership values are n-truples of real numbers in the unit interval [0, 1] ordered in increasing order, called n-dimensional intervals. The set of n-dimensional intervals is denoted by L n ([0, 1]). This paper aims to investigate the class of functions on L n ([0, 1]) which are continuous and strictly decreasing, called n-dimensional strict fuzzy negations. In particular, investigate the class of representable n-dimensional strict fuzzy negations, i.e., n-dimensional strict fuzzy negation which are determined by strict fuzzy negation. The main properties of strict fuzzy negations on [0, 1] are preserved by representable strict fuzzy negations on L n ([0, 1]). In addition, the conjugate obtained by action of an n-dimensional automorphism on an n-dimensional strict fuzzy negation provides a method to obtain other n-dimensional strict fuzzy negations, in which the properties of the original one are preserved, as well as the Fodor's characterization theorem.