@Charlie
this derivation of the Riemann tensor shows how to see the Riemann tensor as a 2-form with two additional indices naturally, and combining with my post on connections
yesterday, you can see that if $\mathbf{A} = A^{\rho} \mathbf{e}_{\rho}$ lives in the same space as your $x^{\mu}$, then it has all space-time indices,