Can the perturbation of the spin connection $\delta \omega^{a}_{b \mu}$ be determined by perturbing the first structure equation $de^{a} = -\omega^{a}_{b}\wedge e^{b}$ ?

Question

I am trying to uderstand the behavior of Dirac spinors in terms of the scalar perturbations of the FRW metric and I was wondering:  

Can the perturbation of the spin connection $\delta \omega^{a}_{b \mu}$ be determined by perturbing the first structure equation $de^{a} = -\omega^{a}_{b}\wedge e^{b}$ ?  More explicitly, can i use the following

$ d\left(\delta e^{a}\right) = -\delta\omega^{a}_{b}\wedge \bar{e}^{b}-\bar{\omega}^{a}_{b}\wedge \delta e^{b}$

?

In the above, the background quantities are being denoted with a bar.