A Ricci flow (M, g_t) on an n-dimensional Riemannian manifold $M$ is an intrinsic geometric flow. A family of smoothly embedded submanifolds $M_t, g_E) of a fixed Euclidean space E = R^n is called an extrinsic representation in R^n of (M, g_t) if there exists a smooth one-parameter family of isometries (M_t, g_E)\rightarrow (M, g_t). When does such a representation exist? We show that for a Ricci flow initialized by a compact surface of revolution immersed in R^n, such a representation can always be constructed. Moreover, we provide a comprehensive framework for this construction, allowing us to exhibit, in particular, the first explicit extrinsic representations in R^4 of the Ricci flows initialized by toroidal surfaces of revolution immersed in R^3.