So all lines that are skew to a fixed line become the points of a 4-dimensional affine space. If the ground field F is a proper skew field then there is no point model like the Klein quadric. However, it is still possible to endow the set of all lines that are skew to a fixed line with the structure of an affine space. Thus the construction from above can be carried out irrespective of the existence of the Klein quadric. From a local point of view this affine space may serve as a substitute for the Klein quadric.
Similar constructions are possible for higher dimensions.
Baer Subspaces within Segre Manifolds, Result. On Isomorphisms of Grassmann Spaces, Mitt. Affine Spaces within Projective Spaces, Result. Best viewed with your own eyes.