Robots operating in dynamic and shared environments benefit from anticipating contact before it occurs. We present GenTact-Prox, a fully 3D-printed artificial skin that integrates tactile and proximity sensing for contact detection and anticipation. The artificial skin platform is modular in design, procedurally generated to fit any robot morphology, and can cover the whole body of a robot. The skin achieved detection ranges of up to 18 cm during evaluation. To characterize how robots perceive nearby space through this skin, we introduce a data-driven framework for mapping the Perisensory Space -- the body-centric volume of space around the robot where sensors provide actionable information for contact anticipation. We demonstrate this approach on a Franka Research 3 robot equipped with five GenTact-Prox units, enabling online object-aware operation and contact prediction.