Generally a determinant function is a function $\delta:V^n\rightarrow F$ where $V$ is a vector space and $F$ is a field such that the map is multilinear and alternating.
We also know that the the the function $\text{det}$ is a determinant-function. Where with $\text{det}$ I mean the "canonical/standard" determinant function over a matrix. We know $\text{det}(E)=1$
I want to show there exists a determinant function $q$ such that for a given scalar $\delta$ and a given basis $X=a_1,\ldots,a_n$ of $V$ we have $q(X)=\delta$
I have tried to show it but all my attempts involved roots and I am trying to avoid this.