My question is about computing the first Chern class from the trace of curvature 2-form matrix via the Chern–Weil theory. First Chern class is given by $$ c_{1}=\left[{\frac {i}{2\pi }}\operatorname {tr} \Omega \right] ...(1)$$ where the curvature 2-form matrix is defined as $ \Omega =d\omega +{\frac {1}{2}}[\omega ,\omega ] \, \ $ or $ \, \ {\Omega^a}_b = d{\omega^a}_b + {\omega^a}_i \wedge {\omega^i}_b \, \ ... \ (2) $.
It is obvious that the trace of a real 2-form valued matrix $\Omega$ is always zero. So, I'm confused when trying to compute the first Chern class from the trace of curvature 2-form matrix. $ \mathbb{CP}^1$ case is simple since we only have one component of $\Omega$ and trace is simply the Kähler 2-form.
For example, I have computed $\Omega$ matrix by using equation (2) for $ \mathbb{CP}^2$. Trace of $\Omega$ matrix is zero as expected. But we know that $ c_1(\mathbb{CP}^2)$ is not zero. I know that there is another way to compute first Chern class of $ \mathbb{CP}^2$ which includes cohomology groups etc. but I need to specifically solve/understand this trace issue.
I am using real coordinates such as $r, \phi, \theta $ and $ \psi$ for explicit calculations. It is possible to write the Fubini-Study metric by using 2 complex coordinates but I couldn't figure out how to compute the complexified $\Omega$ and how to implement it in computing the first Chern class.