An invariant that determines membership in an element of $Gr(2,\mathbb{C}^3)$

Question

Consider two basis vectors for a two-dimensional subspace of $\mathbb{C}^3,$ $v_1=(z_1,z_2,z_3),v_2=(z'_1,z'_2,z'_3)$. I am looking for a geometric or algebraic invariant that determines if another vector $v_3\in$ span$\{v_1,v_2\}$ in a generalized version of this solution for $Gr(1,\mathbb{C}^n)$: For a vector $v=(z_1,z_2,...,z_n)$, the ratios $\frac{|z_i|}{|z_j|}$ and $angle(z_i,z_j)$ (i.e. the oriented angle between components) are invariant under scalar multiples of $v$. Therefore, a 1-dimensional subspace of $\mathbb{C}^n$ is determined by ${n \choose 2}$ values for ratios of lengths and ${n \choose 2}$ angles.