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.
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.