We build a microscopic model to study the intralayer and interlayer superexchange due to electrons hopping in chromium trihalides (, Cl, Br, and I). In evaluating the superexchange, we identify the relevant intermediate excitations in the hopping. In our study, we find that the intermediate hole-pair excitations in the orbitals on ion play a crucial role in mediating various types of exchange interactions. In particular, the interlayer antiferromagnetic exchange may be realized by the hole-pair-mediated superexchange. Interestingly, we also find that these virtual hopping processes compete with each other, leading to weak intralayer ferromagnetic exchange. In addition, we also study the spin-orbit coupling effects on the superexchange and investigate the Dzyaloshinskii-Moriya interaction. Finally, we extract the microscopic model parameters from density functional theory for analyzing the exchange interactions in a monolayer .