  We define and compute explicitly the classical limit of the realizations of
$W_n$ appearing as hamiltonian structures of generalized KdV hierarchies. The
classical limit is obtained by taking the commutative limit of the ring of
pseudodifferential operators. These algebras---denoted $w_n$---have free field
realizations in which the generators are given by the elementary symmetric
polynomials in the free fields. We compute the algebras explicitly and we show
that they are all reductions of a new algebra $w_{\rm KP}$, which is proposed
as the universal classical $W$-algebra for the $w_n$ series. As a deformation
of this algebra we also obtain $w_{1+\infty}$, the classical limit of
$W_{1+\infty}$.
