A fibration of R^n by oriented copies of R^p is called skew if no two p-planes intersect nor contain parallel directions. We discuss some interesting features of skew fibrations, and we exhibit a deformation retract from the space of fibrations of R^3 by skew oriented lines to its subspace of Hopf fibrations. As a corollary of the proof we obtain Gluck and Warner's classification of great circle fibrations of S^3. We conclude with a discussion of skew fibrations in higher dimensions, including some surprising connections to the Hurwitz-Radon function and to vector fields on spheres.