Robotic dexterous grasping is a challenging problem due to the high degree of freedom (DoF) and complex contacts of multi-fingered robotic hands. Existing deep reinforcement learning (DRL) based methods leverage human demonstrations to reduce sample complexity due to the high dimensional action space with dexterous grasping. However, less attention has been paid to hand-object interaction representations for high-level generalization. In this paper, we propose a novel geometric and spatial hand-object interaction representation, named DexRep, to capture dynamic object shape features and the spatial relations between hands and objects during grasping. DexRep comprises Occupancy Feature for rough shapes within sensing range by moving hands, Surface Feature for changing hand-object surface distances, and Local-Geo Feature for local geometric surface features most related to potential contacts. Based on the new representation, we propose a dexterous deep reinforcement learning method to learn a generalizable grasping policy DexRepNet. Experimental results show that our method outperforms baselines using existing representations for robotic grasping dramatically both in grasp success rate and convergence speed. It achieves a 93% grasping success rate on seen objects and higher than 80% grasping success rates on diverse objects of unseen categories in both simulation and real-world experiments.