finally in order to show that a non-empty set is a subspace you do not need to prove that it contains unfortunately some instructors give this wrong idea to some students.
for part (b), the basis you found is wrong! you should at least see that none of the elements of the set that you think is a basis basically belongs to anyway, a basis of
has only one element. for example
for part (c) first see that an matrix with real (or complex) entries, is antisymmetric iff for all and for all can you
see the general form of ? if not, try to do it for n = 3 first. now it should be easy to show that a basis for antisymmetric matrices has elements. what are they?