It doesn't store energy in the sense that all that work you did compressing it doesn't increase its internal energy. It just has the energy it started with, floating around in the atmosphere. Now, some of that energy can be converted to work, if the air is adiabatically expanded, but particularly at high compression it's not large compared to the energy that went into compressing the air (and that was dissipated as heat when the hot compressed air was cooled.)