I just started to read about tensor products and tensors and I understand that a tensor product $V \otimes W$ is a space used to replace bilinear maps $V \times W \to U$ with linear maps $V \otimes W \to U$. Now I have tried to solve a few simple exercises and I realize that I have no idea how to actually "work" with those tensor products/tensors.
Let's look at $u_1 = \begin{pmatrix} 1\\ 1\\ 0 \end{pmatrix}, u_2 = \begin{pmatrix} 0\\ -1\\ 1 \end{pmatrix}, u_3 = \begin{pmatrix} 1\\ 0\\ -1 \end{pmatrix}$ as a basis of $V = \mathbb{R}^3$. Let $G$ be a tensor in $V^* \otimes V^*$, given by $G(x,y) = x_1 y_3 + x_2 y_2,$ where $x = (x_1, x_2, x_3), y = (y_1, y_2, y_3)$ in $\mathbb{R}^3$. Express $G$ in terms of the basis $(u_i^* \otimes u_j^*)$ of $V^* \otimes V^*$.
Looking at the solution, this exercise should be really easy to solve, as we can write $G = e_1^* \otimes e_3^* + e_2^* \otimes e_2^*.$ Why can we write $G$ in this way? Maybe I am lacking some basic knowledge of tensors, because without the solution, I would have no idea how to even approach this exercise and even with the solution, I have no idea why it is true.