A skyrmion state in a non-centrosymmetric helimagnet displays topologically protected spin textures with profound technological implications for high density information storage, ultrafast spintronics, and effective microwave devices. Usually, its equilibrium state in a bulk helimagnet occurs only over a very restricted magnetic-field--temperature phase space and often in the low temperature region near the magnetic transition temperature Tc. We have expanded and enhanced the skyrmion phase region from the small range of 55-58.5 K to 5-300 K in single-crystalline Cu2OSeO3 by pressures up to 42.1 GPa through a series of phase transitions from the cubic P2(_1)3, through orthorhombic P2(_1)2(_1)2(_1) and monoclinic P2(_1), and finally to the triclinic P1 phase, using our newly developed ultrasensitive high-pressure magnetization technique. The results are in agreement with our Ginzburg-Landau free energy analyses, showing that pressures tend to stabilize the skyrmion states and at higher temperatures. The observations also indicate that the skyrmion state can be achieved at higher temperatures in various crystal symmetries, suggesting the insensitivity of skyrmions to the underlying crystal lattices and thus the possible more ubiquitous presence of skyrmions in helimagnets.