I would like to prove
$$K=\frac12g^{jk}R^{i}_{ijk}$$
I know that $$K(p)=I(R(v,w)w,v)$$
for an orthonormal basis $(v,w)$ of $T_pS$. (I want to consider a $2$-dimensional manifold, i.e. a surface)
Using a parametrisation $F:U\rightarrow V$, I find
$$ \begin{align} I(R(v,w)w,v) &=I(R^{l}_{ijk}v^iw^jw^k\partial_lF,v^m\partial_mF)\\ &=R^{l}_{ijk}v^iw^jw^kv^mg_{lm}\\ \end{align}$$
I think I have to use $(v,w)=(\frac{\partial_1 F}{\|\partial_1 F\|},\frac{\partial_2 F}{\|\partial_2 F\|})$, if it was orthogonal in the first place... So I do not really know how to move on.
Thanks for your attention. I'd be very grateful for some help!