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: 
The book *Algèbre linéaire et Géométrie élémentaire* by Jean Dieudonné is a highly formal presentation of elementary geometry using linear algebra. It's the way the Bourbaki school might have taught geometry in high school.

Two books called *Linear Algebra and Geometry*, one by Kostrikin and Manin, the other by Shafarevich and Remizov, apply linear algebra to geometry, but don't limit themselves to revisiting elementary topics.

Artin's book *Algebra* uses many examples from geometry to illustrate the theory (groups, etc.).

Finally, Volume 2 of the book *Fundamentals of Mathematics*, edited by Behnke and Bachmann, examines geometry from various axiomatic perspectives which, while modern, rigorous, and at times quite abstract, are perhaps closer in spirit to Euclid than to an exposition of geometry through linear algebra.
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.