I recently stumbled upon the fascinating online series on linear algebra by the youtube channel 3blue1brown and have been hunting for geometrical interpretations and intuitive explanations for every single expression i had encountered in linear algebra.The n by n determinant however is not arrived at by a definition that involves geometry at all; instead it stems from a set of clearly defined properties that it alone is capable of satisfying. Is it possible to show that the formula for an n by n determinant is also the factor by which the volume equivalent in n dimensions is scaled by? if so how does one go about such a feat? and does the recursive nature of the definition have any significance or is it just a compact way of expression that is merely coincidental?
PS: if possible, please direct me to additional resources and please dont do the mathematical equivalent of pulling out a lightsaber to pick a lock and end up giving an answer that i'm incapable of neither comprehending nor accepting.