Given four vectors $\vec x\_0$, $\vec x\_1$, $\vec x\_2$, and $\vec x\_3$, I would like to write the expression $$\det (\vec x\_1, \vec x\_2, \vec x\_3) - \det (\vec x\_0, \vec x\_2, \vec x\_3) + \det (\vec x\_0, \vec x\_1, \vec x\_3)-\det (\vec x\_0, \vec x\_1, \vec x\_2)$$ more compactly, using sigma notation. For example, $$\sum\_{k=0}^3 (-1)^k \det (\vec x\_0, \vec x\_1, \vec x\_2, \vec x\_3){\rm (but\space k)}.$$

Is there a notation for what I have written as ${\rm (but\space k)}$?