Construction of Hadamard Matrices of Order $n!$ I'm trying to get a hand on Hadamard matrices of order $n!$, with $n>3$. Payley's construction says that there is a Hadamard matrix for $q+1$, with $q$ being a prime power.
Since
$$
n!-1 \bmod 4 = 3
$$
construction 1  has to be chosen:

If $q$ is congruent to $3 (\bmod 4)$ [and $Q$ is the corresponding Jacobsthal matrix] then
  $$    H=I+\begin{bmatrix} 0 & j^T\\ -j & Q\end{bmatrix} $$
  is a Hadamard matrix of size $q + 1$. Here $j$ is the all-1 column vector of length $q$ and $I$ is the $(q+1)×(q+1)$ identity matrix. The matrix $H$ is a skew Hadamard matrix, which means it satisfies $H+H^T = 2I$.

The problem is that the number of primes among $n!-1$ is restricted (see A002982). I checked the values of $n!-1$ given by Wolfram|Alpha w.r.t. be a prime power, without success, so Payley's construction won't work for all $n$. 

Is there a general way to get the matrices, or is it case by case different?
I haven't yet looked into Williamson's construction nor Turyn type constructions. Would it be worth a closer look (sure it would, but) concerning my problem? Where can I find their constructions?

PS for the interested reader: I've found a nice compilation of Hadamard matrices here: http://neilsloane.com/hadamard/
 A: I don't think a general construction for Hadamard matrices of order $n!$ is known. The knowledge about general construction methods for Hadamard matrices is quite sparse, the basic ones (see also the Wikipedia article) are:
1) If $n$ is a multiple of $4$ such that $n-1$ is a prime power or $n/2 - 1$ is a prime power $\equiv 1\pmod{4}$, then there exists a Hadamard Matrix of order $n$ (Paley).
2) If $n$ is a multiple of $4$ such that there exists a Hadamard Matrix of order $n/2$, then there exists a Hadamard Matrix of order $n$ (Sylvester).
The Hadamard conjecture states that for all multiples $n$ of $4$ there is a Hadamard matrix of order $n$.
The above constructions do not cover all these $n$, the smallest case not covered is $n = 92$.
There are more specialized constructions and a few computer constructions, such that the smallest open case is $n = 668$ nowadays.
EDIT:
I have just checked that for $n\in\{13,26,44,52,63,67,70,77,85\}$ a Hadamard matrix of order $n!$ cannot be constructed only by a combination of the Paley/Sylvester construction above. So in these cases, one had to check more specialized constructions like Williamsons' one.
