Due to the considerable fading arising from the long-distance transmission between satellites and terrestrial terminals as well as the complex geographic environments on the earth, aerial relays have been designed to facilitate the information delivery between the satellites and terrestrial terminals when reliable direct data transmissions cannot be achieved. In this paper, a cooperative satellite-aerial-terrestrial network (CSATN) system including a satellite (S), an aerial relay (R), and a terrestrial receiver (D) is considered, while decode-and-forward relay scheme is adopted. Specifically, inspired by the beam shape of the transmission signal from S, we model the operation space of R as a circular truncated cone to accurately reflect the practical large-scale transmission scenarios, the height of which is decided by the upper and lower limitations of the operation height of R. Thus, considering the randomness of R and D and employing geometric probability theory, the exact and asymptotic outage probability (OP) of S-R, S-D, and R-D links is studied, and then the end-to-end outage performance of CSATN is investigated accordingly by using approximation method. Finally, numerical results are presented to validate our proposed analysis.