Dimensionality is one of the most important properties of complex physical systems. However, only recently this concept has been considered in the context of complex networks. In this paper we further develop the previously introduced definitions of dimension in complex networks by presenting a new method to characterize the dimensionality of individual nodes. The methodology consists in obtaining patterns of dimensionality at different scales for each node, which can be used to detect regions with distinct dimensional structures as well as borders. We also apply this technique to power grid networks, showing, quantitatively, that the continental European power grid is substantially more planar than the network covering the western states of US, which present topological dimension higher than their intrinsic embedding space dimension. Local dimension also successfully revealed how distinct regions of network topologies spreads along the degrees of freedom when it is embedded in a metric space.