How to efficiently solve a large system of equations

This question is a follow-up of my last question.

Summary

The user Henrik Schumacher gave an excellent answer about how to store equations efficiently and solve the associated homogeneous system by using Nullspace. Here, I write again his code. The function cf generate the matrix of coefficients of the linear equations

cf = With[{Part = CompileGetElement},
Compile[{{λ, _Real, 2}},
Block[{f, mm},
mm = Length[λ];
f = Table[λ[[i, 1]] λ[[j, 2]] - λ[[i, 2]] λ[[j, 1]], {i, 1, mm}, {j, 1, mm}];
Flatten@Table[
f[[i, j]] f[[k, m]] f[[n, p]], {i, 1, mm}, {j, 1, mm}, {k, 1, mm}, {m, 1, mm}, {n, 1, mm}, {p, 1, mm}
]
],
CompilationTarget -> "C",
RuntimeAttributes -> {Listable},
Parallelization -> True,
RuntimeOptions -> "Speed"
]
];


You can then generate 250 equations in one shoot in only 30ms

A = cf[
Divide[
DeveloperToPackedArray[
N[RandomInteger[{-30, 30}, {250, 5, 2}]]],
DeveloperToPackedArray[N[RandomInteger[{1, 4}, {250, 5}]]]
]
]; // AbsoluteTiming // First


0.03101

Now we can solve the system numerically with NullSpace in only 7 seconds

nullspace = NullSpace[A]; // AbsoluteTiming // First


7.59116

Question

I find the method suggested by Henrik Schumacher very useful but I am wondering whether I can improve it more. Indeed, the quantity f defined in the body of the function cf is a two rank tensor, anti-symmetric in its indexes. So, in the Table output there a lot of redundant terms. For example, for mm=5, the code generates a matrix with $$5^6=15625$$ entries. If you consider that f is a rank-2 anti-symmetric matrix in 5 dimension, then only $$(10)^3 = 1000$$ terms should be computed, which would be a good achievement.

I would like to use the symmetry property of f to further improve the code and speed up the whole process. Indeed, if you only add an extra factor of f[[y,u]] in the Table inside cf, i.e.

     Flatten@Table[
f[[i, j]] f[[k, m]] f[[n, p]]f[[y,u]], {i, 1, mm}, {j, 1, mm}, {k, 1, mm}, {m, 1, mm}, {n, 1, mm}, {p, 1, mm},{y,1,mm},{u,1,mm}
]


then the time to generate the matrix A gets multiplied by a factor of 10 and Mathematica's kernel aborts when evaluating Nullspace.

It is even worse if I add two extra factors of f, in which case Mathematica takes 50 seconds to generate the matrix and i have no hope to compute the NullSpace.

How can I include anti-symmetry of f` to simplify the system?