Let $n$ be a natural number. Let $k$ be an element of $\{1, \ldots , n\}$. For each j in $\{1, \ldots , n\}$, I want to find a bijection $f_j$ from $S_{n-1}$ to $\{\sigma \in S_n : \sigma(k) = j \}$, where $S_m$ denotes the symmetric group of order $m!$. This bijection should (hopefully) have the property that for any $\sigma \in S_{n-1}$, $\text{sgn}(f_j(\sigma)) = (-1)^{j+k}\text{sgn}(\sigma)$.
I have attempted this several times and can't seem to find a bijection that works in every situation. Also, I'm not sure how one would prove the property about the signs of the permutations. Any help with one or both of these issues would be greatly appreciated.