(This is just a comment on Anton's answer. I originally posted it as two comments, but there were too many typos.)

I agree with Anton's central point that in the end you have to decide what notation and conventions to use and the correct definition of sectional curvature is determined by the criteria given by Anton.

However, every reference I know defines
$$
R(X,Y)Z = ([\nabla_X, \nabla_Y] - \nabla_{[X,Y]})Z.
$$
In that case, the wikipedia definition is correct.

However, what gets confusing is that many authors want, with respect to an orthonormal frame $e_1, \dots, e_n$, the sectional curvature of the plane spanned by $e_i$ and $e_j$ to be given by $R_{ijij}$ and not $R_{ijji}$. This can be pulled off by defining
$$
R^i{}_{jkl}e_i = R(e_k, e_l)e_j
$$
and $R_{ijkl} = g_{ip}R^p{}_{jkl}$.

ADDED: It appears that I am wrong about the convention above being universal. However, I do believe it is the more widely used convention. In any case, Anton's advice is sound.