Using the combined theoretical approaches, the structural, electronic, and magnetic properties of the two-dimensional (2D) $1T$ phase of monolayer ${\mathrm{RhO}}_{2}$ were studied. Density-functional theory plus $U$ calculation indicate that $1T\text{\ensuremath{-}}{\mathrm{RhO}}_{2}$ favors a ferromagnetic metallic phase with an out-of-plane magnetization, and it can be achieved by exfoliation from the layered bulk oxides containing monolayer $1T\text{\ensuremath{-}}{\mathrm{RhO}}_{2}$. Monte Carlo simulation shows that the ferromagnetic phase is stable below the Curie temperature of 73.9 K based on the Heisenberg Hamiltonian model. Magnetic anisotropy energy and its cause were further investigated to understand the microscopic origin of the 2D magnetization. Our results indicate that spin-orbit coupling interaction between Rh $4d$ stabilizes perpendicular magnetic anisotropy, resulting in the out-of-plane magnetization, over in-plane magnetic anisotropy. In addition, mechanical tensile strain can strengthen perpendicular magnetic anisotropy by increasing positive contribution of the interaction to the magnetic anisotropy energy.