# [Jelly], 19 bytes

 LŒ!s€2Ṣ€QḅL_LịFHP€S

[Try it online!][TIO-je9yvhnf]

### Alternate version, 15 bytes, postdates challenge

 LŒ!s€2Ṣ€QœịHP€S

Jelly *finally* got n-dimensional array indexing.

[Try it online!][TIO-je9z0j2i]

### How it works

 LŒ!s€2Ṣ€QœiHP€S Main link. Argument: M (matrix / 2D array)

 L Take the length, yielding 2n.
 Œ! Generate all permutations of [1, ..., 2n].
 s€2 Split each permutation into pairs.
 Ṣ€ Sort the pair arrays.
 Q Unique; deduplicate the array of pair arrays.
 This avoids dividing by n! at the end.
 H Halve; yield M, with all of its elements divided by 2.
 This avoids dividing by 2**n at the end.
 œị At-index (n-dimensional); take each pair of indices [i, j] and
 yield M[i][j].
 P€ Take the product the results corresponding the same permutation.
 S Take the sum of the products.

The 19-byte version works in a similar fashion; it just has to implement `œị` itself.

 ...ḅL_LịFH... Return value: Array of arrays of index pairs. Argument: M

 L Length; yield 2n.
 ḅ Convert each pair of indices [i, j] from base 2n to integer,
 yielding ((2n)i + j).
 _L Subtract 2n, yielding ((2n)(i - 1) + j).
 This is necessary because indexing is 1-based in Jelly, so the
 index pair [1, 1] must map to index 1.
 F Yield M, flattened.
 ị Take the indices to the left and get the element at these indices
 from the array to the right.
 H Halve; divide all retrieved elements by 2.

[Jelly]: https://github.com/DennisMitchell/jelly
[TIO-je9yvhnf]: https://tio.run/##y0rNyan8/9/n6CTF4kdNa4we7lwEpAIf7mj1ifd5uLvbzSMAyA/@f7gdSEX@/x/NpRAdbaCjYKJnGqujEA2kdBQMYmN14MLmIMIMSQFQAChhpGcIpPVMIGJmMAEdBUM9I4RBeiZgAYSRhnrGIKMMYcJ6BiDCEiQGNcoQxDYCmWcOVwQ2xRwkbwiXMoHoArOAhhhD5MEGghWBpYxALGOQnYZ6FhB5S6iBUF3GIK4pkv0Qu8AGgnWBpUBcI5AXYgE "Jelly – Try It Online"
[TIO-je9z0j2i]: https://tio.run/##TY89DsIwDIV3ThH2yKqTtKU3YGAAMaEoIwvqxsTKwsBWbsDMBRAj6kHKRYLt9G@Jn9@zPyunY11fYty0zfL8u75M935S2bWP7nNfb0nu4/dG5RCjXyjvM60c5EErT0WrLAQ92iU/xWyADAoMIFVwySsGQysEM4HAiTEhESyjcLAh46dir0cha8O8chwSSsk5jpFLW6IIYlMuQBmSyLCyfBNhlfKqB/Zbltt8dj/dEqBsScSt4S@EPw "Jelly – Try It Online"