Why people use the Gram-Schmidt process instead of just chosing the standard basis I really can't find a reason for going through all the work of the Gram-Schmidt method to make a new orthogonal basis $B'$ given an old basis $B$.
If I want to change to an orthogonal basis, the most simple solution to me is just the standard basis.

Yes in $\mathbb R^n$ the Gram-Schmidt procedure need not be particularly helpful, since there's often an obvious orthonormal basis.
However, if we start with a different vector space, such as the polynomials $\mathbb R[x]$ over the interval $[0,1]$ with an inner product, the first few elements of an orthonormal basis is not obvious and Gram-Schmidt is helpful.
