How to create a polygon on an affine space?

Question

If I had an affine space with a basis $\{a_0, a_1, a_2\}$, I could use these points to either create a triangle or select other three points $\{c_0, c_1, c_2\}$ on the plane to create the triangle $\triangle(c_0, c_1, c_2)$ which, I believe, the set of points on and in the triangle is $\{t_0c_0 + t_1c_1 + t_2c_2 : t_0 + t_1 + t_2 = 1 \; \text{and} \; t_0, t_1, t_2 \geq 0\}$. My question is in what way could I go into defining the set of points on and in other polygons given a basis $\{a_0, a_1, a_3\}$ and vertices $\{c_0, c_1, \ldots, c_n\}$, such that any three are affinely independent.