Question: I'm searching for some references that deal with topics from "elementary geometry" analysing them from a "higher" perspective (for example, abstract algebra, linear algebra, and so on).

Answer: 
Kaplansky's *Linear algebra and geometry* does a nice readable overview of the connection with linear algebra. Artin's *Geometric algebra* goes further in depth! but is a little less fun to read. Then there are Kaplansky's references to Coxeter's books which I imagine are great, but I haven't had the time to read them yet.
Try Jürgen Richter-Gebert: *Perspectives on Projective Geometry*. It shows the classical material in a modern way, and is written in an excellent pedagogical style.