The projection viewpoint for a rotation matrix using dot products is totally valid, but I think the column viewpoint is more intuitive.
A 3x3 matrix can be viewed as something that acts on the standard unit basis vectors. The action on the standard unit basis vectors can be read out by looking at the columns. (A matrix times the ith standard basis vector is just the ith column.) Since the transform is linear, all the other points in space are carried along with the action as well. This is the "linear map" viewpoint of a rotation matrix.
A 3x3 matrix can be viewed as something that acts on the standard unit basis vectors. The action on the standard unit basis vectors can be read out by looking at the columns. (A matrix times the ith standard basis vector is just the ith column.) Since the transform is linear, all the other points in space are carried along with the action as well. This is the "linear map" viewpoint of a rotation matrix.
I show an example here https://biro.ai/we-learned-the-wrong-way-to-matrix-multiply/ where this viewpoint lets you instantly see what a rotation matrix does by looking at the columns.