For given basis $(\vec{v_k})$ of Euclidean space $\mathbb{E}^n$ find representation of vector $\vec{w}$ expressed by dot product $<\vec{v_k}, \vec{w}>$. In other words coefficients of $\vec{w}$ have to be expressed by dot product.
All I could think of is to use Gram-Schmidt process to form orthonormal basis expressed in terms of $\vec{v_k}$ and then easily just take dot products of $\vec{w}$ with each of these basis vectors. But this form is far from the expressions of $<\vec{v_k}, \vec{w}>$ and so on.
Is there another way which actually gives coefficients discribed by dot products mentioned before?