About eigenvectors of sorted skew-symmetric Toeplitz matrices

Question

I was playing around with Toeplitz matrices, specifically skew-symmetric Toeplitz matrices. So the diagonal is a zero, every diagonal above (resp. below) the main diagonal is a negative of its counterpart below (resp. above). The image on the left below is an example. Then I additionally constrained such a matrix to have a descendingly sorted first row (so largest values are close to the diagonal) - this looks like the image on the right below.

Skewify[mat_] := 
 Module[{n, m}, {n, m} = Dimensions@mat; 
  Table[Which[j == k, 0, j < k, 1, j > k, -1], {j, 1, m}, {k, 1, n}]*
   mat]

GraphicsRow[
 With[{mm = RandomReal[{-10, 10}, 20]}, {MatrixPlot[
    Skewify[N@ToeplitzMatrix@mm]], 
   MatrixPlot[Skewify[N@ToeplitzMatrix@Reverse@Sort@mm]]}]]

Above is the Mathematica code for that. I am looking for some explanation of the following phenomenon for the eigenvectors of such matrices: They seem to sit on ellipses/circles/spirals around the origin.

{v1, v3, v5, v7} = 
  Eigenvectors[
      Skewify[N@
        ToeplitzMatrix@
         Reverse@Sort@RandomReal[{-10, 10}, 100]]][[#]] & /@ {1, 3, 5,
     7};
GraphicsGrid@
 Partition[
  ListPlot[(Tooltip[{Re[#1], Im[#1]}] &) /@ # , 
     AspectRatio -> 1] & /@ {v1, v3, v5, v7}, 2]

Can someone point to me what could be going on here? I don't even have a clue where to start. If it is too big to answer here, any references/papers would be welcome too. (I am not a mathematician, just a classically trained engineer but I can take a stab at it).

Note: The above image is generated assuming the entries in the first row of the matrix are chosen uniformly at random from a symmetric interval (the RandomReal[{-20, 20}, 100] part); one can get other interesting plots by restricting these to be positive (e.g RandomReal[{0, 20}, 100]), negative and so on. Here is an example with RandomReal[{-20, 0}, 100]).

PROGRESS:

With an intention to do something like what is done for circulant matrices, for a $n\times n$ skew-symmetric Toeplitz matrix whose first row runs as $ \{0, a_2, a_3, \dots, a_{n}\}$ I have derived the eigenvector equation to be:

$$ \sum \limits_{k=m+1}^n a_{k-m+1} v_k - \sum \limits_{k=1}^{m-1} a_{m-k+1}v_k = \lambda v_m $$ for $m=1, 2, \dots, n$ assuming $v$ is an eigenvevtor and $v_m$ is its $m$-th component.

But now stuck at this point. How do I use the fact that $a_2 \geq a_3 \geq \dots \geq a_n$?