The characteristic polynomial of this family of matrices I'm looking at the following family of $n\times n$ matrices. The entries are 0 everywhere except above and below the diagonal. Above it takes values from $1 \to n-1$ and below from $ -n +1 \to -1$. Example when $n=4$:
$\left[
\begin{matrix}
0 & 1 & 0 & 0 \\
-3 & 0 & 2 & 0 \\
0 & -2 & 0 & 3 \\
0 & 0 & -1 & 0 \\
\end{matrix} \right]
$
How can I find the characteristic polynomial of these matrices in general?

Let $D=\operatorname{diag}(1,i,i^2,\ldots,i^{n-1})=\operatorname{diag}(1,i,-1,-i,1,i,-1,-i,\ldots)$. Then $D^{-1}AD=iK$ where
$$
K=\pmatrix{
0&1\\
n-1&0&2\\
 &n-2&\ddots&\ddots\\
 & &\ddots&\ddots&\ddots\\
 & &      &\ddots&0     &n-1\\
 & &      & & 1 &0}.
$$
is known as the Kac matrix. The spectrum of $K$ is given by
$$
\{-n+1,\,-n+3,\,-n+5,\ldots,\,n-5,\,n-3,\,n-1\}.
$$
Hence the eigenvalues of $A$ are
$$
\{(-n+1)i,\,(-n+3)i,\,(-n+5)i,\ldots,\,(n-5)i,\,(n-3)i,\,(n-1)i\}
$$
and the characteristic polynomial of $A$ is
\begin{cases}
x&\text{when $n=1$,}\\
[x^2+(n-1)^2][x^2+(n-3)^2]\cdots(x^2+4^2)(x^2+2^2)x&\text{when $n\ge3$ is odd,}\\
[x^2+(n-1)^2][x^2+(n-3)^2]\cdots(x^2+3^2)(x^2+1)&\text{when $n$ is even.}\\
\end{cases}
