The coupling of the spin of electrons to their motional state lies at the heart of recently discovered topological phases of matter. Here we create and detect spin-orbit coupling in an atomic Fermi gas, a highly controllable form of quantum degenerate matter. We directly reveal the spin-orbit gap via spin-injection spectroscopy, which characterizes the energy-momentum dispersion and spin composition of the quantum states. For energies within the spin-orbit gap, the system acts as a spin diode. We also create a spin-orbit coupled lattice and probe its spinful band structure, which features additional spin gaps and a fully gapped spectrum. In the presence of s-wave interactions, such systems should display induced p-wave pairing, topological superfluidity, and Majorana edge states.National Science Foundation (U.S.)United States. Defense Advanced Research Projects Agency. Optical Lattice Emulator ProgramUnited States. Army Research Office. Multidisciplinary University Research InitiativeUnited States. Air Force Office of Scientific Research. Multidisciplinary University Research InitiativeUnited States. Office of Naval Research. Young Investigator ProgramUnited States. Defense Advanced Research Projects Agency. Young Faculty AwardUnited States. Air Force Office of Scientific Research. Presidential Early Career Award for Scientists and EngineersDavid & Lucile Packard FoundationEngineering and Physical Sciences Research Council (Grant EP/I010580/1)