Typically permutations are discussed in the context of finite sets. I seem to recall, however, at least one source saying that every bijection of a set onto itself is a permutation. That was in the context of discussing the transformations of the Euclidean plane.
Is it a generally accepted definition to say that a permutation is any bijection of a set onto itself?
The answer is "no" -- there is no universally accepted standard for the term "permutation" on an infinite set. See discussions here and here. Two natural choices are the definition presented (all bijections), and the set of all bijections with all but finitely many points fixed.