A wrinkle formation mechanism with cutaneous aging is addressed through a mechanical calculation of linear buckling. Skin is divided into five mechanically distinct layers in this study. In general, the outer layer is stiffer than the inner layer, so buckling occurs in the outer layer against the uniform compression caused by muscle contraction. This buckling damages the skin and affects the formation of permanent wrinkles. We propose a multistage buckling theory for evaluation of the wrinkle property, namely, the specific wrinkle size and critical strain in three stages. The specific wrinkle size is derived as the wavelength of the minimum-buckling mode for infinite-length skin. A sensitivity analysis is carried out to investigate the effect of age-related changes of the mechanical parameters on the wrinkle property. We employ some aging hypotheses and prepare two sets of mechanical parameters, one for young and one for aged skin. The numerical results show that the buckling mode switch from Stage I to Stage II is the main reason why pronounced wrinkles suddenly appear in aged skin.