What would be a basis of $L^2(\Omega )$

Question

Let $(\Omega ,\mathcal F,P)$ a probability space and $$L^2(\Omega )=\{random\ variable\ X\mid \mathbb E[X^2]<\infty \}$$ is a vector space. What would be a basis of $L^2(\Omega )$ ? I also know that $\left<X,Y\right>\longmapsto \mathbb E[XY]$ is a semi-inner product. Could we find an orthonormal basis of $L^2(\Omega )$ ?