Yes, for the "quasi-classical" case(i.e. for the case when the $W$-algebra is commutative, which occurs when
the level is either infinite or critical) it was defined by Drinfeld and Sokolov long
time ago; you can look at Section 4 of http://arxiv.org/PS_cache/math/pdf/0305/0305216v1.pdf
for a good review.

For the "quantum" case (i.e. for arbitrary level) it was studied by Feigin and Frenkel, but I am not sure what the right reference is; you can look for example at Section 4
of http://arxiv.org/PS_cache/hep-th/pdf/9408/9408109v1.pdf, but there should be more modern references. In fact, the main tool in the work of Feigin and Frenkel is the screening operators, which describe the $W$-algebra explicitly as a subalgebra of (the vertex operator algebra associated to) the Heisenberg algebra (where the embedding to the Heisenberg algebra is the Miura transformation).

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