I'm trying to think through the hierarchy of mathematical objects leading to an orthonormal basis. A space may be filled by vectors and a vector may be thought of as a linear combination of components in a basis. But, if a basis is not orthonormal, is each vector constituting said basis not itself a linear combination of components in an orthonormal basis?
Yes, you can make a non-orthonormal basis from an orthonormal basis. Having an orthonormal basis is a stronger condition than just having a basis. You might want to look into the Gram-Schmidt process for some insight on the relation of orthonormal basis and non-orthonormal basis.