as a complex vector space, the dimension is indeed 6, and NOT 12. the number 1 (= 1 + 0i) is in fact a basis for over the field . i do not wish to seem pedantic about this, but you are completely wrong, and liable to confuse the thread-starter. HallsofIvy is completely correct that the underlying field does affect the dimension, but post #7 does not assert that the original question is to be interpreted as the matrices in question being a "real vector space".
this is contextually clear from the subsequent question:
1) The set of 3*2 with real entries (.....is it a subspace?)
this is sort of a "trick question" because the subset is closed under addition, but it is NOT closed under scalar multiplication, for such a (non-zero) matrix A, the matrix iA is not in the set.