Can one define a cross product for functions?

Question

The dot product $c = \sum_i a_i b_i$ can be easily be generalized for continuous functions like $$ c = \int_{-\infty}^{\infty} a(x) b(x) d x $$ But can one also generalize the cross product $c_{ij} = (a_ib_j-a_jb_i)$ to continuous functions in a similar way?

Answer 1

let $$c(x,y)=a(x)b(y)-a(y)b(x)$$