Solve linear system with $A_{i,j} = \langle e_i, e_j\rangle^2$, edges of a tetrahedron

Question

I have six vectors in $e_i\in\mathbb{R}^3$ that are the edges of a tetrahedron. Let us consider now the linear equation system $Ax=b$ with $$ A_{i,j} = \langle e_i, e_j\rangle^2,\\ b_i = \langle e_i, e_i\rangle. $$ The $3\times 3$-system for a triangle has the exact solution $$x_1 = \frac{\langle e_2, e_3\rangle}{\langle e_1\times e_2, e_1\times e_3\rangle}. $$ (The other components likewise.)

Any ideas on how to approach this for a tetrahedron?