We present a novel method for constructing neurally implemented spatial representations that we show to be useful for building models of spatial cognition. This method represents continuous (i.e., real-valued) spaces using neurons, and identifies a set of operations for manipulating these representations. Specifically, we use "fractional binding" to construct "spatial semantic pointers" (SSPs) that we use to generate and manipulate representations of spatial maps encoding the positions of objects. We show how these representations can be transformed to answer queries about the location and identities of objects, move the relative or global position of items, and answer queries about regions of space, among other things. We demonstrate that the neural implementation in spiking networks of SSPs have similar accuracy and capacity as the mathematical ideal.