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But I wanted to know if there is a way of thinking the cross product "in terms of" the dot product.

That is because the dot product can be generalized to an inner product, and from R^3 to an arbitrary inner vector space (and Hilbert space). In that process of generalization, where does the cross product fit?

(1) I mean, is there a way (at least in R^3) of calculating a cross product using only the dot product?

(2) Moreover, I want to know, the definition of euclidean space R^n, is the vector space along with the usual inner product (dot product), or with any other inner product also is considered an eucliden space?

Thanks!