The first part is easy. You must show that if A and B are anti-symmetric matrices and r is a number, A+ B and rA are also anti-symmetric. And that follows from the definitions of matrix addition and multiplication of a matrix by a number.
As for the dimension, the space of all n by n matrices has dimension $\displaystyle n^2$ because you can choose any of the $\displaystyle n^2$ entries arbitrarily. For anti-symmetric matrices, you know that numbers on the diagonal must be 0. How many numbers are there on the diagonal? How many other numbers does that leave? And once you have chosen the numbers above the diagonal, the numbers below must be their negatives. How many numbers are there above the diagonal?