Prove that the eigenvalues of skew-Hermitian matrices are purely imaginary

Let $x$ an eigenvector of $A$ associated to the eigenvalue $\lambda$ then

$$\langle Ax,x\rangle=\overline \lambda||x||^2=\langle x,A^*x\rangle=-\langle x,Ax\rangle=-\lambda\langle x,x\rangle$$ so $$\overline\lambda=-\lambda$$ hence $\lambda$ is pure imaginary complex.