Strain-tunable Dzyaloshinskii-Moriya interaction and skyrmions in two-dimensional Janus <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>Cr</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:msub><mml:mi>X</mml:mi><mml:mn>3</mml:mn></mml:msub><mml:msub><mml:mi>Y</mml:mi><mml:mn>3</mml:mn></mml:msub></mml:mrow></mml:math> ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>X</mml:mi><mml:mo>,</mml:mo><mml:mo> </mml:mo><mml:mi>Y</mml:mi></mml:math> = Cl, …

Recently, great effort has been devoted to the search for two-dimensional (2D) ferromagnetic materials with inherent strong Dzyaloshinskii--Moriya interaction (DMI). Here, through a first-principles approach, we systematically investigate effect of biaxial strain on DMI, Heisenberg exchange interaction, and magnetic anisotropy energy (MAE) Janus Cr$_{2}$X$_{3}$Y$_{3}$ (X, Y = Cl, Br, I, X $\neq$ Y) monolayers. Both DMI MAE can be significantly enhanced by tensile strain, while reversal chirality in Cr$_{2}$Cl$_{3}$Br$_{3}$ switch from off-plane in-plane Cr$_{2}$I$_{3}$Cl$_{3}$ are induced compressive $2\%$. Microscopically, associated mainly large spin--orbit coupling heavy nonmagnetic halogen atoms rather than that Cr atoms. In particular, peculiar transition is explained competition between direct superexchange interactions. Micromagnetic simulations show small external field 65~mT stabilizes skyrmion diameter 9.8~nm Cr$_2$I$_3$Cl$_3$ monolayer. Our results will provide guidance further research skyrmions 2D materials, as well basis potential applications spintronic devices.