show/hide this revision's text 3 Hopefully improved LaTeX, and other minor changes.

Here's a bit of history on cross products that, if not directly useful, will hopefully provide some context. They were defined in "Beno Eckmann, Stetige Losungen linearer Gleichungssysteme, Comment. Math. Helv. 15(1943)" as follows: An $r$-fold cross product on a real vector space $V$ of dimension $n$ with inner product $g$ is a continuous map \[ P : \underbrace{V \times \cdots \times V}_r \to V \] satisfying

1) $P$ is skew;
\[ g\big( P(v_1, \ldots ,v_r),v_i\big) = 0 \ , \ 1 \leq i \leq r \ , \] 2) $P$ respects $g$;
\[ g\big(P(v_1, \ldots ,v_r),P(v_1, \ldots ,v_r)\big) = \det g(v_i,v_j) \ . \]

These were classified by Eckmann and Whitehead (see "George W. Whitehead, Note on cross-sections in Stiefel manifolds, Comment. Math. Helv.37 (1962/1963)") using algebro-topological methods. They were later also classified by Brown and Gray (see "Robert B. Brown and Alfred Gray, Vector cross products, Comment. Math. Helv. 42 (1967)") where those authors included an extra axiom: $P$ has to be multilinear. This extra axiom makes no significant difference to the classification. The classification theorem is:

An $r$-fold cross product on a real vector space $V^n$ exists if and only if we have one of

$\bullet$ $n$ even, $r=1$,
$\bullet$ $n=7$, $r=2$,
$\bullet$ $n=8$, $r=3$,
$\bullet$ $n$ arbitrary, $r=n-1$.

The proof of Brown and Gray uses Hurwitz' structure theorem for composition algebras. If you add a dimension to $V$ you can define a composition algebra, and vice versa. Their paper is my favourite reference. They actually consider a more general situation where the bilinear form is indefinite, that leads to more cross products (but only in the four cases above). They even work with any field of characteristic not $2$.

Using the standard inner product on $V = \mathbb{R}^n$ (which you are implicitly using by referring to $\ast$), your cross product is the last one on the list (it is a cross product in this sense, I was wrong to say otherwise earlier). It is normally just called the volume form, with the appropriate identifications made. I don't know about the earlier history of that particular case, or whether this was thought of as a cross product prior to the papers I've mentioned.

As Ryan points out, in two dimensions there is a $1$-fold cross product, and this is rotation by $90^{\circ}$. That's because $1$-fold cross products are the same as complex structures. The usual $2$-fold cross product on $\mathbb{R}^3$ is a special case of the volume form, and fits into the fourth case of the classification.

I cannot miss the chance to briefly mention the role of cross products in geometry. One can define a cross product on the tangent bundle of a Riemannian manifold. Kaehler manifolds are those with (parallel) $1$-fold cross products, and the cross products in seven and eight dimensions correspond to the exceptional holonomies.
show/hide this revision's text 2 Corrected definition, added comment on bilinear form

Here's a bit of history on cross products that, if not directly useful, will hopefully provide some context. They were defined in "Beno Eckmann, Stetige Losungen linearer Gleichungssysteme, Comment. Math. Helv. 15(1943)" as follows: An $r$-fold cross product on a real vector space $V$ of dimension $n$ with inner product $g$ is a continuous map \[ P : V^{\otimes r} \underbrace{V \times \cdots \times V}_r \to V \] satisfying

1) $P$ is skew;
\[ g\big( P(v_1, \ldots ,v_r),v_i\big) = 0 \ , \ 1 \leq i \leq r \ , \] 2) $P$ respects $g$;
\[ g\big(P(v_1, \ldots ,v_r),P(v_1, \ldots ,v_r)\big) = \det g(v_i,v_j) \ . \]

Since $P$ is skew and we have an inner product we can think of a cross product as a map $\Lambda^rV \to V$.

These were classified by Eckmann and Whitehead (see "George W. Whitehead, Note on cross-sections in Stiefel manifolds, Comment. Math. Helv.37 (1962/1963)") using algebro-topological methods. They were later also classified by Brown and Gray (see "Robert B. Brown and Alfred Gray, Vector cross products, Comment. Math. Helv. 42 (1967)") where those authors included an extra axiom: $P$ has to be linearmultilinear. This extra axiom makes no significant difference to the classification. The classification theorem is:

An $r$-fold cross product on a real vector space $V^n$ exists if and only if we have one of

$\bullet$ $n$ even, $r=1$,
$\bullet$ $n=7$, $r=2$,
$\bullet$ $n=8$, $r=3$,
$\bullet$ $n$ arbitrary, $r=n-1$.

The proof of Brown and Gray uses Hurwitz' structure theorem for composition algebras. If you add a dimension to $V$ you can define a composition algebra, and vice versa. Their paper is my favourite reference. They actually consider a more general situation where the bilinear form is indefinite, that leads to more cross products (but only in the four cases above). They even work with any field of characteristic not $2$.

Using the standard inner product on $V = \mathbb{R}^n$ (which you are implicitly using by referring to $\ast$), your cross product is the last one on the list (it is a cross product in this sense, I was wrong to say otherwise earlier). It is normally just called the volume form, with the appropriate identifications made. I don't know about the earlier history of that particular case, or whether this was thought of as a cross product prior to the papers I've mentioned.

As Ryan points out, in two dimensions there is a $1$-fold cross product, and this is rotation by $90^{\circ}$. That's because $1$-fold cross products are the same as complex structures. The usual $2$-fold product on $\mathbb{R}^3$ is a special case of the volume form.

I cannot miss the chance to briefly mention the role of cross products in geometry. One can define a cross product on the tangent bundle of a Riemannian manifold. Kaehler manifolds are those with (parallel) $1$-fold cross products, and the cross products in seven and eight dimensions correspond to the exceptional holonomies.
show/hide this revision's text 1

Here's a bit of history on cross products that, if not directly useful, will hopefully provide some context. They were defined in "Beno Eckmann, Stetige Losungen linearer Gleichungssysteme, Comment. Math. Helv. 15(1943)" as follows: An $r$-fold cross product on a real vector space $V$ of dimension $n$ with inner product $g$ is a continuous map \[ P : V^{\otimes r} \to V \] satisfying

1) $P$ is skew;
\[ g\big( P(v_1, \ldots ,v_r),v_i\big) = 0 \ , \ 1 \leq i \leq r \ , \] 2) $P$ respects $g$;
\[ g\big(P(v_1, \ldots ,v_r),P(v_1, \ldots ,v_r)\big) = \det g(v_i,v_j) \ . \]

Since $P$ is skew and we have an inner product we can think of a cross product as a map $\Lambda^rV \to V$. These were classified by Eckmann and Whitehead (see "George W. Whitehead, Note on cross-sections in Stiefel manifolds, Comment. Math. Helv.37 (1962/1963)") using algebro-topological methods. They were later also classified by Brown and Gray (see "Robert B. Brown and Alfred Gray, Vector cross products, Comment. Math. Helv. 42 (1967)") where those authors included an extra axiom: $P$ has to be linear. This extra axiom makes no significant difference. The classification theorem is:

An $r$-fold cross product on a real vector space $V^n$ exists if and only if we have one of

$\bullet$ $n$ even, $r=1$,
$\bullet$ $n=7$, $r=2$,
$\bullet$ $n=8$, $r=3$,
$\bullet$ $n$ arbitrary, $r=n-1$.

The proof of Brown and Gray uses Hurwitz' structure theorem for composition algebras. If you add a dimension to $V$ you can define a composition algebra, and vice versa.

Using the standard inner product on $V = \mathbb{R}^n$ (which you are implicitly using by referring to $\ast$), your cross product is the last one on the list (it is a cross product in this sense, I was wrong to say otherwise earlier). It is normally just called the volume form, with the appropriate identifications made. I don't know about the earlier history of that particular case, or whether this was thought of as a cross product prior to the papers I've mentioned.

As Ryan points out, in two dimensions there is a $1$-fold cross product, and this is rotation by $90^{\circ}$. That's because $1$-fold cross products are the same as complex structures. The usual $2$-fold product on $\mathbb{R}^3$ is a special case of the volume form.

I cannot miss the chance to briefly mention the role of cross products in geometry. One can define a cross product on the tangent bundle of a Riemannian manifold. Kaehler manifolds are those with (parallel) $1$-fold cross products, and the cross products in seven and eight dimensions correspond to the exceptional holonomies.