We theoretically study the stability of magnetic skyrmion crystals (SkXs) in chiral cubic antiferromagnets with the EuPtSi hosting unconventional nanometric SkXs in mind. By performing numerical simulations for a minimal effective spin model with the long-range Dzyaloshinskii-Moriya interaction and the multiple-spin interaction arising from itinerant nature of electrons, we clarify two key ingredients for the emergence of the SkXs in EuPtSi: One is the low-symmetric ordering vectors that lead to the field-direction sensitive SkXs in an external magnetic field, while the other is the synergy between spin-charge and spin-orbit couplings in itinerant magnets that results in the nanometric SkXs. The present study will not only provide the microscopic mechanism of the SkXs in EuPtSi but also guide for a search of further materials hosting short-period SkXs.