Symbolic second order derivative of real antisymmetric matrix in Mathematica

I would like to calculate the second order derivative of a function with respect to a matrix coefficient symbolically. Below is the first order derivative of S with respect to the coefficient of x. For getting the answer what I have done is I have expressed the S as the function of only coefficients.

myD[S , x_[n_, s_]] := D[S[x[n, s]], x[n, s]];
   

Here S is a function of the big matrix ‘x’. But as I said before I need to find the derivative of S with respect to the coefficient of x.

myD[myD[S, x_[n_, s_]], x_[t_, r_]] := 
  D[S[x[n, s]], {x[n, s], 2}];

Above code is what I have written for second derivative and of course it is not correct as what I need is something as below equation.

Is there a best way to define that?
$$sum_{nu<mu}sum_{beta<alpha}x_{ialpha}x_{beta j}x_{kmu}x_{nuell}frac{partial^{2}S}{partial x_{numu}x_{betaalpha}}$$
This is how it participate.
Definition:
$$frac{partial^{2}S}{partial x_{numu}x_{betaalpha}}=frac{partial}{partial x_{numu}}left(frac{partial S}{partial x_{betaalpha}}right)$$

If we have,
$$K=x_{lalpha}x_{beta k}x_{inu}x_{mu j}$$

$$frac{partial SK_{left(betaalpharight),left(upsilonmuright)}}{dx_{upsilonmu}}=Sfrac{partial K_{left(betaalpharight),left(upsilonmuright)}}{dx_{upsilonmu}}+Kfrac{partial S}{dx_{upsilonmu}}$$

$$frac{partial^{2}SK_{left(alphabetaright),left(muupsilonright)}}{dx_{betaalpha}x_{upsilonmu}} =frac{partial}{dx_{betaalpha}}left(frac{partial SK_{left(betaalpharight),left(upsilonmuright)}}{dx_{upsilonmu}}right)
=frac{partial}{dx_{betaalpha}}left(Sfrac{partial K_{left(betaalpharight),left(upsilonmuright)}}{dx_{upsilonmu}}+Kfrac{partial S}{dx_{upsilonmu}}right)
=left(Sfrac{partial^{2}K_{left(betaalpharight),left(upsilonmuright)}}{dx_{betaalpha}x_{upsilonmu}}+frac{partial K_{left(betaalpharight),left(upsilonmuright)}}{dx_{upsilonmu}}frac{partial S}{dx_{betaalpha}}+Kfrac{partial^{2}S}{dx_{betaalpha}x_{upsilonmu}}+frac{partial K_{left(betaalpharight),left(upsilonmuright)}}{dx_{betaalpha}}frac{partial S}{dx_{upsilonmu}}right)
$$

provided D is the dimension of the matrix.
Also,
$$ frac{partial x_{lj}}{dx_{betaalpha}}=delta_{j,alpha}delta_{l,beta}-delta_{j,beta}delta_{l,alpha}$$

EDITED
But when I put the rule of differentiation of real antisymmetric matrix as in the above equation by the following code, it is showing error.

myD[x_[k_, l_], 
   x_[v_, g_]] := [Delta][k, v] [Delta][l, g] - [Delta][k, 
     g] [Delta][l, v];

Clear[myD, x]
myD[myD[a_, b__], c__] := myD[a, b, c]

Format[HoldPattern@myD[a_, b__]] := TraditionalForm@HoldForm@D[a, b]

Format[x[a_, b_]] := Subscript[x, a, b]

myD[myD[S, x[ν, μ]], x[α, β]]

myD[a_ + b_, c_] := myD[a, c] + myD[b, c]

myD[a_ b_, c_] := a myD[b, c] + b myD[a, c]

Format[x[a_, b_]] := Subscript[x, a, b]

myD[myD[S K, x[β, α]], x[ν, μ]]