I realize that Axler's Linear Algebra Done Right is a great textbook, and the removal of determinants is often pedagogically sound. However, I was doing some problems on a practice entrance exam, and I was surprised by the amount of times solutions would involve having to apply determinants to geometric situations.
Therefore, my question is this: Where I can learn more about determinants, preferably without having to relearn all of linear algebra from another textbook?
For an easy introduction to determinants there are a lot of online resources. See for example
Very basic intro to determinant
Another intro to determinant with graphical representation
For a very nice graphic representation have a look at this youtube video.
Hope this helps!