Let $u_1,\dots,u_n$ be vectors in an inner-product space and define $a_{ij} = \langle u_i , u_j \rangle$. So $$ A=(a_{ij}) = \begin{pmatrix} u_1^{\dagger} \\ \vdots\\ u_n^{\dagger} \end{pmatrix} \begin{pmatrix} u_1 & \cdots & u_n \end{pmatrix} = U^{\dagger} U, $$ where $U$ is the matrix of column vectors $u$ stacked side-by-side. But then $$ x^{\dagger} A x = \langle Ux,Ux \rangle \geqslant 0, $$ with equality only if $Ux=0$, i.e. $x=0$ or the matrix $U$ is singular, which occurs precisely when the $u_i$ are not independent.
Therefore $A$ is a positive operator, and hence (by Sylvester's criterion) all the principal minors are positive: $$ a_{11}>0, \\ a_{11} a_{22}-\lvert a_{12} \rvert^2 > 0, \\ a_{11}a_{22}a_{33} - (a_{11}\lvert a_{23}\rvert^2 +a_{22}\lvert a_{31}\rvert^2+a_{33}\lvert a_{12}\rvert^2 ) + 2\Re(a_{12}a_{23}a_{31}) > 0, \\ \vdots $$ with equality only if there is linear dependence in the system. The first is trivial, the second is Cauchy–Schwarz, the third and on are unfamiliar.
My question is, are the subsequent inequalities just consequences of Cauchy–Schwarz itself? Do they have any particular use? And are they a known result?
Have a look here, https://en.wikipedia.org/wiki/Gramian_matrix. You'll find the relationship to be $$ || u_1 \wedge \ldots \wedge u_n ||^2 $$