Michael Hardy's conjecture is definitely not rubbish.

I know of at least two definitions that apply to more than three dimensions. For both definitions, however, the resulting cross product is a vector subspace rather than a vector. (See Notes 1 and 2)

As Mr Hardy notes, in three dimensions the conventional cross product of vectors a and b is a vector normal to a and b with length (ab)sin(Theta), where Theta is the angle between a and b. It has the property that its inner product (a x b)*c with any vector c is the volume of the parallelepiped having edges a, b and c. The vector c is uniquely defined and complements the {a, b} space(Note 3).

In more than three dimensions, however, the normal to two vectors is not unique. For dimensions n > 3, the cross product may be defined to be the n-2 dimensional subspace normal to the two vectors. Alternately, the cross product may be defined to be the n-m dimensional subspace normal to m vectors, m>2. This later possibility is pursued here.

Let the vectors w1, w2, w3, … , wm span an m-dimensional subspace Sm in the n-dimensional vector space V, where m = 2, 3, …, n-1. The cross product of the vectors w1, w2, w3, … , wm is then defined as cm:

```
cm = w1 x w2 x … x wm
= |s1||s2| … |sm| Cn-m,
```

where

```
sm = wm - [Sm-1]*wm
Sm = { s1, s2, … , sm}.
```

Here Cn-m is the complementary space (Note 4)of Sm in V, defined as all vectors normal to Sm. The vector sm is normal to Sm-1 and [Sm-1]*wm is the projection(5) of wm onto Sm-1. The angle Thetam between wm and Sm-1*wm is given by Sm= |wm||sm|(Thetam), where |sm| and |wm| are the lengths of vectors sm and wm.

If m = n-1 then Cn-m = C1, a vector normal to the other n-1 vectors. If m = n, the product |s1||s2| … |sm|is the volume of the parallelepiped having edges w1, w2, w3, … , wm. This definition agrees with the conventional one for n = 2, 3 but, unlike the conventional definition, yields a null vector if any of the w1, w2, w3, … , wm are dependent.

In both definition, the cross product is a subspace rather than a vector. I have not yet explored the first definition or the relation of either to the Hodge star.

NOTES:
(1) For reference, the subspace of m (non-parallel) vectors in a larger n-dimensional space is just the collection of points that may be reached by adding any combination of m vectors with multiplicative coefficients. Any m non-parallel vectors "span" an m-dimensional subspace, m<=n.
(2) In the above, the suffixes m and n are integers, lower case characters are vectors, and upper case characters are subspaces; the notation {x, y} indicates the space spanned by the vectors x and y; [S] indicates a set of orthonormal vectors in the space S; the symbol * indicates inner product.
(3) in the sense that any point in V may be reached by a linear combination of a, b and c.
(4) Halmos, Paul R., Finite Dimensional Vector Spaces, Second Edition, 1958, D. Van Nostrand Company, Inc., Princeton, N. J., p29.
(5) P is a projection if PPx = Px for any vector x. (ibid. p73.) It can be shown that (Px)*(x – Px) = 0, so that x – Px is perpendicular to Px.