We study the formation of a glassy skin at the air interface of drying polymer solutions. We introduce a simple approximation, which is valid for most diffusion problems, and which allows us to derive analytical relationships for the polymer concentration as a function of time. We show that the approximate results differ by less than 15% from those obtained by numerically solving the diffusion equation. We use the approximation to study skin formation in evaporating solutions. We focus on the influence of variations of the mutual diffusion coefficient with concentration, when the latter decreases sharply at high concentrations, as observed in the vicinity of the glass transition. We show that the skin thickness depends very strongly on the exponent characterising the decrease of the diffusion coefficient, in contrast to the polymer volume fraction at the interface, which varies only slightly with the exponent.