Skyrmions are localized solitonic spin textures with protected topology, which are promising as information carriers in ultradense and energy-efficient logic and memory devices. Recently, magnetic skyrmions have been observed in magnetic thin films, and are stabilized by the extrinsic interfacial Dzyaloshinskii-Moriya interaction (DMI) and/or external magnetic fields. The specific effects in magnetic monolayer materials have not been thoroughly studied. Here, we investigate the intrinsic magnetic skyrmions in a family of monolayer Janus van der Waals magnets, MnSTe, MnSeTe, VSeTe, and MnSSe, by the first-principles calculations combined with the micromagnetic simulations. The monolayer Janus MnSTe, MnSeTe, and VSeTe with out-of-plane geometric asymmetry and strong spin-orbit coupling have a large intrinsic DMI, which could stabilize a sub-50-nm intrinsic skyrmions in monolayer MnSTe and MnSeTe at zero magnetic field. Whereas a monolayer VSeTe with an in-plane easy axis forms a magnetic domain rather than skyrmions. Moreover, the size and shape of skyrmions can be tuned by an external magnetic field. Therefore, our work motivates a vista for seeking intrinsic skyrmions in atomic-scale magnets.