Given statement is
Note: In this research paper all calculation is under field of characteristic $3$ unless specified.
$\pmb{proposition:}$ Let $K$ be a field of arbitrary characteristics. Let $X$ be a skew-symmetric $(n \times n)-$ matrix over $K$ and $H=(h_{ij})$ be skew-symmetric with $$h_{ij} = \begin{cases} 0, & i=j \\[2ex] 1, & i >j \\[2ex] -1 & i<j \\ \end{cases}$$ Then there is $Q$ such that $X=QHQ^T.$
Lemma: Let $K$ be a field of arbitrary characterstic. Let S be a skew-symmetric $(n \times n)-$ matrix over $K$ and $\hat H=(\hat h_{ij})$ with
$$\hat h_{ij} = \begin{cases} 0, & i=j \\[2ex] 1, & i =j+1 , i \leq n-1 \\[2ex] -1 & i=j-1, i\geq2 \\ \end{cases}$$
Then there is $P$ such that $S=P\hat HP^T.$
Proof: We apply above lemma twice. first $S=H$ and get $P$ such that $$S=H=P\hat HP^T$$ from this we get $$\hat H= P^{-1} H(P^{-1})^T.$$ Next with $S=X$ and get $P_1$ such that
$$S=X=P_1 \hat H (P_1)^T=P_1P^{-1}H(P^{-1})^T(P_1)^T $$
Now we put $$Q=P_1P^{-1}.$$
My doubts (1) In the given lemma I did't get all the entries of any order matrix. That is the matrix in the lemma is not well defined I think!
(2) As all the calculation in this paper is over modulo $3$ So can I take for particular $n=3$ numbering of rows as $0, 1 ,2 $ and same for column $0,1,2$ if yes then also i did't get all entries of the matrix $H=\hat h_{ij}$
So, If matrix in the lemma is not correct the how we can define it properly??
and then we will be able of understand the proof(then it seems easy!)
Also, I read this answer for the proof Whether a nondegenerate skew-symmetric matrix is congruent to the matrix $\begin{bmatrix} 0 & I_{\ell} \\ -I_{\ell} & 0 \end{bmatrix}$
but I want to to understand the above procedure.
So please any help is well appreciated