Atomic hydrogen and oxygen are the dominant species in the Martian exosphere. Atomic hydrogen is essentially produced from the dissociation of H2O, whereas, hot oxygen atoms are populated by non-thermal processes such as the dissociative recombination of O2+ with electrons in the Martian ionosphere. The study of these species helps to understand the evolution of the Martian atmosphere and more specifically the history of water on Mars. The Emirates Mars Ultraviolet Spectrometer (EMUS), one of the primary instruments onboard the Emirates Mars Mission (EMM), has been observing atomic hydrogen and oxygen in the Martian exosphere over the Mars Year 36. We present the analysis of the cross-exospheric observations by the EMUS for hydrogen Lyman series and oxygen 130.4 nm emissions and their seasonal variability. The EMUS cross-exospheric observations cover the tangent altitude starting from 130 km to more than 35,000 km above the disk, with most of the observations below 25,000 km. The observations show that when Mars moved from perihelion to aphelion, the hydrogen emission line intensities increase by an order of magnitude or more whereas, for oxygen, it is an increment by a factor of about 2 at larger altitudes. Based on these observations, we also discuss the retrieval of densities, temperature, and the estimation of escape fluxes of hydrogen and oxygen species by applying 3D hydrogen ballistic corona and 3D Monte Carlo particle transport models, respectively.