Questions about ODE with constant coefficients

When we solve the characteristic equation, it is inevitable that sometimes we may have complex roots, and we simply use the Euler formula to expand the solution and take $e^{at}cosbt, e^{at}sinbt$ as the basic solution. But do these two basic solutions still independent from the other $(n-2)$ basic solutions? Is there any possible convenient way to prove using the Wronski determinant?

Yes, they are independent of the others. The proof is the following (informally):

First, you prove that your 2 complex conjugate solutions $z$ and $\bar{z}$ are independent from the other solutions. This implies that they span a linearly independent subspace of your space of solutions.

Then, any linear combination of $z$ and $\bar{z}$ will belong to that same subspace, and will therefore be independent of all the other solutions. In other words, you can perform local changes of basis in that subspace without losing the linear independence from the other solutions.

In particular, the Euler formula provides a change of basis for that subspace, and gives you two new basis functions which are independent from each other (and by the above reasoning also independent from all the other solutions).

Cheers!