Ascending and descending Morse complexes, defined by the critical points and integral lines of a scalar field f defined on a manifold domain D, induce a subdivision of D into regions of uniform gradient flow, and thus provide a compact description of the morphology of f on D. Here, we propose a dimension independent representation for the ascending and descending Morse complexes, and a data structure which assumes a discrete representation of the field as a simplicial mesh, that we call the incidence-based data structure. We present algorithms for building such data structure for 2D and 3D scalar fields, which make use of a watershed approach to compute the cells of the Morse decompositions. We describe generalization operators for Morse complexes in arbitrary dimensions, we discuss their effect and present results of our implementation of their 2D and 3D instances both on the Morse complexes and on the incidence-based data structure.