Let $\mathcal{R}$ be a ring and let $v^0,\ldots,v^{k-1}\in\mathcal{R}^m$ with $m \geq k$. Suppose we wish to find $w\in Span(v^0,\ldots,v^{k-1})$ such that $k-1$ specified coordinates of $w$ vanish (say $w[j_0] = \ldots = w_[j_{k-2}] = 0$). Then the following determinantal construction for $w$ does the trick:

$w = -\sum_{i=1}^k (-1)^k\cdot\det\begin{pmatrix}{v^0}[j_0]&{v^0}[j_1]&\ldots&{v^0}[j_{k-2}]\\\\{v^1}[j_0]&{v^1}[j_1]&\ldots&{v^1}[j_{k-2}]\\\\\vdots&\vdots&\ddots&\vdots\\\\\widehat{v^i[j_0]}&\widehat{v^i[j_1]}&\ldots&\widehat{v^i[j_{k-2}]}\\\\\vdots&\vdots&\ddots&\vdots\\\\{v^k}[j_0]&{v^k}[j_1]&\ldots&{v^k}[j_{k-2}]\end{pmatrix}\cdot v^i$

The individual coordinates of $w$ also have a nice determinantal form (which follows from the above expression):

$w[i] = \det\begin{pmatrix}v^0[i]&v^0[j_0]&v^0[j_1]&\ldots&v^0[j_{k-2}]\\\\v^1[i]&v^1[j_0]&v^1[j_1]&\ldots&v^1[j_{k-2}]\\\\\vdots&\vdots&\vdots&\ddots&\vdots\\\\v^{k-1}[i]&v^{k-1}[j_0]&v^{k-1}[j_1]&\ldots&v^{k-1}[j_{k-2}]\end{pmatrix}$

For example, suppose we start with the 3 vectors $v^0 = \langle 2,3,5,7,11,13\rangle$, $v^1 = \langle17,19,23,29,31,37\rangle$ and $v^2 = \langle 41,43,47,53,59,61\rangle$ and wish to find a vector $w$ in their span whose final 2 coordinates are 0. Then the above formulas give us the following vector:

\begin{align*} w &= \det\begin{pmatrix}31&37\\\\59&61\end{pmatrix}\cdot v^0 - \det\begin{pmatrix}11&13\\\\59&61\end{pmatrix}\cdot v^1 + \det\begin{pmatrix}11&13\\\\31&37\end{pmatrix}\cdot v^2\\\\ &= \left\langle\det\begin{pmatrix}2&11&13\\\\17&31&37\\\\41&59&61\end{pmatrix},\det\begin{pmatrix}3&11&13\\\\19&31&37\\\\43&59&61\end{pmatrix},\det\begin{pmatrix}5&11&13\\\\23&31&37\\\\47&59&61\end{pmatrix},\right.\\\\ &\;\;\;\;\;\;\;\;\;\;\left.\det\begin{pmatrix}7&11&13\\\\29&31&37\\\\53&59&61\end{pmatrix},\det\begin{pmatrix}11&11&13\\\\31&31&37\\\\59&59&61\end{pmatrix},\det\begin{pmatrix}13&11&13\\\\37&31&37\\\\61&59&61\end{pmatrix}\right\rangle\\\\ &= \langle 1212, 1120, 936, 952, 0, 0\rangle \end{align*}

These formulas are easily derived/proved using Cramer's rule and/or other methods involving exterior products (which is how I came up with them when I was trying to construct such a vector), and like Cramer's Rule are rather beautiful, so I would be surprised if they are not already written down/used somewhere. Nevertheless, I don't recall having ever seen such constructions in any Linear Algebra books.

Main Question: Has anyone seen either of the above equivalent formulas before? If so, do they have a name?.

Secondary Question: Whether or not they have a name, has anyone seen these formulas used as part of any other proofs?