It is known that the cardinality of $R$ is equal to the cardinality of $R^2$, $R^3$, etc. But, intuitively these sets have different sizes. A possible way to formalize this intuition is to talk about the dimension of a set, seen as a vector-space over $R$. So $R$ has a dimension of 1, $R^2$ has a dimension of 2, etc.
My question is: what happens when we go to higher cardinalities? Specifically: what is the dimension of $2^{R^2}$, relative to $2^R$? Is there a way to define a basis such that the dimension of $2^{R^k}$ is $k$ over that vector space?