# all possible inner products in $\mathbb R^2$

Suppose $\langle., .\rangle: \mathbb R^2\times \mathbb R^2\to \mathbb R$ is an inner product.

What would be all possible function forms of the inner products, i.e. would all of them have the forms

either $\langle x, y\rangle=ax_1y_1+bx_2y_2$ or $\langle x, y\rangle=ax_1 y_2+b x_2y_1, a,b\in \mathbb R$

or other forms are also possible?