Cross Product in Euclidean Space with infinite Dimensions

By definition the cross product of two vectors is a vector that is perpendicular to the two vectors and has length equal to the area of the parallelogram spanned by the vectors (plus orientation). If the dimension is $<3$ such a vector exists iff the given vectors are parallel. In the 3-space it exists always and is unique. If the dimension is $>3$ it exists but is not unique. In an infinite dim space before defining a cross-product you need to define the length of a vector and the area of a parallelogram.