In a remarkable paper, Peter Hall [On the rate of convergence of normal extremes, J. App. Prob, 16 (1979) 433-439] proved that the supremum norm distance between the distribution function of the normalized maximum of n independent standard normal random variables and the distribution function of the Gumbel law is bounded by 3/ log n. In the present paper we prove that choosing a different set of norming constants that bound can be reduced to 1/ log n. As a consequence, using the asymptotic expansion of a Lambert W type function, we propose new explicit constants for the maxima of normal random variables.