We have mapped the Orion Molecular Ridge (OMC-1 and OMC-2) in the fine-structure lines [C II] 158 mu m and [O I] 63 and 146 mu m. The maps cover a region of 7' x 18' (Delta alpha x Delta delta) and are fully sampled toward the Orion A H II region/molecular cloud interface. The emission maximum near the molecular region S6/FIR4 arises most probably from a photodissociation region (PDR) on the surface of that molecular condensation. The PDR has column densities that are most likely explained by an edge-on geometry. For the emitting material, we find a temperature between 300 and 500 K, a number density of 3 x 10(5) cm(-3), and O-0 and C+ column densities of 1x10(19) and 4x10(18) cm(-2), respectively. In the Orion Bar region, the northern edge of the Orion interface region, and M43 west we find somewhat lower values for the temperature, number density, and column densities. Furthermore, the ratio of the oxygen to carbon column density is lower in this region and approaches a value of 1.7. The lower ratio could be due to the edge-on geometry and the lower temperature of these regions. In the Dark Lane region we find evidence for cooler oxygen partly absorbing the oxygen line emission from the warmer background material. For the cooler molecular material covering the warm medium in the Dark Lane, we estimate an atomic oxygen abundance [O]/[H] on the order of a few 10(-4). In the northern part of OMC-1 the morphology of the [O I] 63 mu m emission is consistent with moderately warm oxygen associated with the dense molecular ridge. From the oxygen emission arising from the ridge, we derive an atomic oxygen abundance [O]/[H] > 10(-5) The [O I] line emission in OMC-2 may arise partly from the inner parts of the molecular cloud, but emission arising exclusively from PDRs cannot be excluded.