In Topology, Geometry, and Physics by Nakahara, the following is given: $$ c_0 (F) = 1\\ c_1(F) = \frac{i}{2\pi}\text{tr}(F)\\ c_2(F) = \frac{1}{2} (\frac{i}{2\pi})^2(\text{tr}(F)\wedge\text{tr}(F) -\text{tr}(F\wedge F)) \\ ... \\ c_k=\frac{i}{2\pi}\det(F)$$
I understand where the expressions for $c_0$ and $c_k$ come from, but from where the other ones come from? Is there an analytical formula for $c_i (F)$ given $1<i<k$?