Do n-by-n traceless matrices form a vector space over the field C of complex numbers? Why or why not? If yes, show a basis and give the dimension of the vector space!

Over the field R of real numbers, I know that they form a vector space:

Tr(A)+Tr(B)=Tr(A)+Tr(B)

TR(c*A)=c*Tr(A)

0 matrix is in the vector space

-A matrix is in

But I don't know what is a basis over real numbers.

And I don't even know if they form a vector space over the field C of complex numbers. I suppose they do, but I am not sure, I can't prove it.

Your help would be appreciated.