The inner product of a complex vector space acts as $\langle \cdot ,\cdot \rangle: V \times V \to \mathbb{C}$, and has a geometric interpretation as returning the angle between two vectors $$ \theta(a,b) = \arccos \left( \sqrt{\frac{\langle a,b \rangle\langle b,a \rangle}{\langle a,a \rangle \langle b,b \rangle}} \right) $$
Is there a name for, geometric interpretation of, and literature surrounding the following object that takes two sets of $n$ vectors and returns the determinant of the matrix of inner products $G: V^n \times V^n \to \mathbb{C}$ $$ G(\{a_1, a_2, \cdots a_n\},\{b_1, b_2, \cdots b_n \} ) = \det\begin{pmatrix} \langle a_1 , b_1 \rangle & \langle a_1 , b_2 \rangle & \cdots & \langle a_1 , b_n \rangle \\ \langle a_2 , b_1 \rangle & \langle a_2 , b_2 \rangle & \cdots & \langle a_2 , b_n \rangle \\ \vdots & \vdots & & \vdots \\ \langle a_n , b_1 \rangle & \langle a_n , b_2 \rangle & \cdots & \langle a_n , b_n \rangle \end{pmatrix} $$ I ask as I noticed it seems a similar angle can be defined $$ \phi(\{a_1, a_2, \cdots a_n\},\{b_1, b_2, \cdots b_n \}) = \arccos \left( \sqrt{\frac{ G(\{a_1, a_2, \cdots a_n\},\{b_1, b_2, \cdots b_n \} ) G(\{b_1, b_2, \cdots b_n\},\{a_1, a_2, \cdots a_n \} )}{G(\{a_1, a_2, \cdots a_n\},\{a_1, a_2, \cdots a_n \} ) G(\{b_1, b_2, \cdots b_n\},\{b_1, b_2, \cdots b_n \} )}} \right) $$ where (I think) $\phi = 0$ if the $a_i$ and $b_j$ have the same span, and $\phi = \pi/2$ if there exists a linear combination of the $a_i$ that is orthogonal to all the $b_j$ (and vice versa).