To write the matrix of a dot product $g:\mathbb{R^2} \times \mathbb{R^2} \to \mathbb{R}$ on a basis $\mathcal{B}=\{v_1, v_2\}$

Question

Let $\mathcal{B}=\{v_1, v_2\}$ be a basis of $\mathbb{R^2}$ such that $v_1 =(3,1)$ and $v_2 =(1,1)$.

Let $g:\mathbb{R^2} \times \mathbb{R^2} \to \mathbb{R}$ a dot product such that:

$g(v_1,v_1)=2$

$g(v_1,v_2)=1$

$g(v_2,v_2)=2$

Write the matrix $C$ associated to $g$ on the basis $\mathcal{B}$.

I am having some doubts about how to do this exercise. If I am right, $C$ is composed in this way: $C= \begin{bmatrix} g(v_1,v_1) & g(v_1,v_2) \\ g(v_2,v_1) & g(v_2,v_2) \\ \end{bmatrix} $, so should I just replace values? $C= \begin{bmatrix} 2 & 1 \\ * & 2 \\ \end{bmatrix} $

If yes, how to find $g(v_2,v_1)$?