in part (a) you have a couple of mistakes: in proving that W is closed under addition that should be changed to also is not in it's in your base field.

finally in order to show that a non-empty set is a subspaceyou do not needto 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?