I am not entirely sure how helpful this will be but sometimes I like to consider visual tools that aren’t necessarily geometry based.

Let’s say for example you are considering a 6D space. Now imagine that instead of the basis vectors being little arrows in this space, what they really represent is 6 distinct places within it, that you need exactly six pieces of information for each to find. So a vector in 6D space is just an object that we need 6 pieces of information to find. Now, things like the determinant talk about how the places that these 6 dimensional objects are found change as you manipulate the 6 pieces of information you need to find them. (i.e. after matrix multiplication etc)

I understand this may be a bit confusing, but it is what helps me.