The Dzyaloshinskii-Moriya interaction (DMI), which only exists in noncentrosymmetric systems, is responsible for the formation of exotic chiral magnetic states. The absence of DMI in most two-dimensional (2D) magnetic materials is due to their intrinsic inversion symmetry. Here, using first-principles calculations, we demonstrate that significant DMI can be obtained in a series of Janus monolayers of manganese dichalcogenides MnXY (X/Y = S, Se, Te, X ≠ Y) in which the difference between X and Y on the opposites sides of Mn breaks the inversion symmetry. In particular, the DMI amplitudes of MnSeTe and MnSTe are comparable to those of state-of-the-art ferromagnet/heavy metal (FM/HM) heterostructures. In addition, by performing Monte Carlo simulations, we find that at low temperatures the ground states of the MnSeTe and MnSTe monolayers can transform from ferromagnetic states with worm-like magnetic domains into the skyrmion states by applying external magnetic field. At increasing temperature, the skyrmion states starts fluctuating above 50 K before an evolution to a completely disordered structure at higher temperature. The present results pave the way for new device concepts utilizing chiral magnetic structures in specially designed 2D ferromagnetic materials.