M(2x2)(R) is a vector space of K.
M =
(a......b)
(-b.....a)
one way of thinking of a basis is it tells us: "how many pieces of information we need to specify". how many numbers (scalars) do you have to know to completely determine the matrix:
$\displaystyle \begin{bmatrix}a&b\\-b&a \end{bmatrix}$?
can you write the matrix as:
a*something + b*(another something)?
what might the "somethings" be?