do you believe in the axiom of choice?

if so, then every vector space has a basis. this includes W. so we may assert we have a basis B for W, and can extend this to a basis C for V.

define W' = span(C\B).

(there are some vector spaces for which such bases have never been adequately described. the usual example is the real numbers as a vector space over Q. R is uncountable, so any basis has to be uncountable, so explicitly listing one is out of the question).