Generalization of a projective plane?

Question

In the area of finite geometry, a projective plane is an incidence structure of points and lines with the following properties:

1. Every two points are incident with a unique line
2. Every two lines are incident with a unique point
3. There are four points, no three colinear

I am trying to figure out a generalization of this concept. Here one finds the following:

Projective planes may be thought of as projective geometries of "geometric" dimension two. Higher dimensional projective geometries can be defined in terms of incidence relations in a manner analogous to the definition of a projective plane

However I cannot find a similar definition. What would be a natural generalization of the above three points to the general dimension $d\geq2$?