  In a remarkable paper, Peter Hall [{\it On the rate of convergence of normal
extremes}, J. App. Prob, {\bf 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.
