Completing a matrix in $SL_n(\mathbb{Z})$

Given $k\in\mathbb{Z}^n$ is there some way to decide when we can construct a matrix $A\in SL_n(\mathbb{Z})$ s.t the first row of $A$ is $k$?

For instance when $n=2$ and $k=(a,b)$ this is just when $a$ and $b$ are relatively prime.

One can construct such a matrix if and only if the gcd of the entries of $k$ is $1$.
The necessity is obvious, the sufficiency not quite so. Given $k$ one can do elementary column operations over $\Bbb Z$ to reduce it to $(1,0,0,\ldots,0)$. Thus there is a matrix $A$ in $\text{GL}_2(\Bbb Z)$ with $kA=(1,0,0,\ldots,0)$ (this is basically the Euclidean algorithm). Then the top row of $A^{-1}$ is $k$.
The determinant of $A^{-1}$ may be $-1$, but I'll leave you to fix that.