By combining the equations $$R_{ijk}{}^l=e^l(R_{ij}e_k)$$ and $$R_{ijkl}= g_{lm}R_{ijk}{}^m$$ on page $197$ and $198$ of Lee's Introduction to Riemannian Manifolds I obtain \begin{equation}\tag{1} R_{ijkl}= g_{lm}R_{ijk}{}^m=g(e_l,e_m)e^m(R_{ij}e_k)=g(e_l,R_{ij}e_k)=g(R_{ij}e_k,e_l). \end{equation} This is consistent with $(2.2.48)$ in Gilkey's $2$nd edition of Invariance theory.
However, on page $34$ of Heat Kernels and Dirac Operators (Getzler et al) and on page $14$ of Roe's Elliptic Operators we have the following definition: $$R_{ijkl}:=g(e_i,R_{kl}e_j)$$ which is different from $(1)$, isn't it?
Edit: AFAIK we can use interchange symmetry and skew symmetry (see Wikipedia) to obtain $$g(e_l,R_{ij}e_k)=g(R_{ij}e_k,e_l)=g(R_{kl}e_i,e_j)=g(e_j,R_{kl}e_i)=-g(e_i,R_{kl}e_j)$$ and this shows that the two definitions are indeed different, right?