Understanding and controlling the electronic structure of nanomaterials is the key to tailoring their use in a wide range of practical applications. Despite this need, many important electronic states are invisible to conventional optical measurements and are typically identified indirectly based on their inferred impact on luminescence properties. This is especially common and important in the study of nanomaterial surfaces and their associated defects. Surface trap states play a crucial role in photophysical processes yet remain remarkably poorly understood. Here we demonstrate for the first time that broadband electronic sum frequency generation (eSFG) microspectroscopy can directly map the optically bright and dark states of nanoparticles, including the elusive below gap states. This new approach is applied to model cadmium selenide (CdSe) quantum dots (QDs), where the energies of interfacial trap states have eluded direct optical characterization for decades. Our eSFG measurements show clear signatures of electronic transitions both above the band gap, which we assign to previously reported one- and two-photon transitions associated with the CdSe core, as well as broad spectral signatures below the bandgap that are attributed to interfacial trap states. In addition to the core states, this analysis reveals two distinct distributions of below gap states providing the first direct optical measurement of both shallow and deep trapping sites on this system. Finally, chemical modification of the surfaces via oxidation results in the relative increase in the signals originating from the interfacial trap states. Overall, our eSFG experiments provide an avenue to directly map the entirety of QD bulk and interfacial electronic structure, which is expected to open up opportunities to study how these materials are grown in situ and how surface states can be controlled to tune functionality.