Cross product - Wikipedia, the free encyclopedia
You can look in the section on generalizations of the cross product into higher dimensions. This will give you a vector that is perpendicular to the subspace spanned by the first 3 vectors. If they are linearly independent, the collection of all 4 should be as well.
Alternatively, guess and check could be popular, you just gotta pick 4 numbers to give you a nonzero determinant that will show your vectors are linearly independent.
Otherwise, yeah, that is probably the best way to do it.