This is confusing because the usual way of writing integrals in differential forms leaves something implicit: the tangent $k$-vector of the manifold of integration, and the notation itself tends to make people confuse $\mathrm dx$---the cotangent basis vector associated with the coordinate $x$---with $dx$---the symbol that tells us we're integrating with respect to $x$.
Suppose you have an integral in differential forms
$$\int_M f(x,y) \, \mathrm dx \wedge \mathrm dy$$
It really means this:
$$\int_M f(x,y) \, (\mathrm dx \wedge \mathrm dy)(T_{(M)}) \, dx \, dy$$
where $T_{(M)}$ is the tangent 2-vector for the manifold of integration $M$. For instance, a common choice of orientation would be that $T_{(M)} = e_x \wedge e_y$. And naturally, $(\mathrm dx \wedge \mathrm dy)(e_x \wedge e_y) = 1$. So the integral would reduce to $\int_M f(x,y) \, dx \, dy$.
This approach naturally generates the relevant surface/volume elements one might need. Suppose you have some surface two-form $\omega$ on a sphere that you want to integrate. You would set up the integral like so:
$$\int_S \omega(\theta, \phi; e_\theta \wedge e_\phi) \, d\theta \, d\phi$$
Note that $e_\theta \wedge e_\phi = r^2 \sin \theta \, \hat \theta \wedge \hat \phi$, and this would be appropriate to use if $\omega$ is written in terms of unit forms instead of $\mathrm d \theta$ and $\mathrm d \phi$.
You can write all of these above in index notation if you like. Just remember that one should always use the tangent $k$-vector associated with the coordinates of integration--$e_\theta \wedge e_\phi$ in that sphere example--though the orientation of that tangent $k$-vector is not necessarily determined by the ordering of the coordinates. This may introduce minus signs, if the problem specifies such.
Most of the time, though, people mean to say that a manifold is oriented the same way as the coordinates are ordered.
I've seen in the literature the notation $C$ with some additional specifications for the contraction maps of all sorts, but the amount of decorations on the symbol $C$ varied depending on the context. See, e.g., A.Gray, Tubes, p.56, where these maps are used in the case of somewhat special tensors, and therefore the notation is simpler.
In general, there is a whole family of uniquely defined maps
$$
C^{(r,s)}_{p,q} \colon \otimes^{r}_{s} V \to \otimes^{r-1}_{s-1} V
$$
which are collectively called tensor contractions ($1 \le p \le r, 1 \le q \le s$).
These maps are uniquely characterized by making the following diagrams commutative:
$$
\require{AMScd}
\begin{CD}
\times^{r}_{s} V @> {P^{(r,s)}_{p,q}} >> \times^{r-1}_{s-1} V\\
@V{\otimes^{r}_{s}}VV @VV{\otimes^{r-1}_{s-1}}V \\
\otimes^{r}_{s} V @>{C^{(r,s)}_{p,q}}>> \otimes^{r-1}_{s-1} V
\end{CD}
$$
Explanations are in order.
Recall that the tensor products $\otimes^{r}_{s} V$ are equipped with the universal maps
$$
\otimes^{r}_{s} \colon \times^{r}_{s} V \to \otimes^{r}_{s} V
$$
where $\times^{r}_{s} V := ( \times^r V) \times (\times^s V^*)$.
Besides that, there is a canonical pairing $P$ between a vector space $V$ and its dual:
$$
P \colon V \times V^* \to \mathbb{R} \colon (v, \omega) \mapsto \omega(v)
$$
Notice that map $P$ is bilinear and can be extended to a family of multilinear maps
$$
P^{(r,s)}_{p,q} \colon \times^{r}_{s} V \to \times^{r-1}_{s-1} V
$$
by the formula:
$$
P^{(r,s)}_{p,q} (v_1, \dots, v_p, \dots, v_r, \omega_1, \dots, \omega_q, \dots, \omega_s) = \omega_q (v_p) (v_1, \dots, \widehat{v_p}, \dots, v_r, \omega_1, \dots, \widehat{\omega_q}, \dots, \omega_s)
$$
where a hat means omission.
Since maps $P^{(r,s)}_{p,q}$ are multilinear, the universal property of the maps $\otimes^{r}_{s}$ implies that there are uniquely defined maps
$$
\tilde{P}^{(r,s)}_{p,q} \colon \otimes^{r}_{s} V \to \times^{r-1}_{s-1} V
$$
and then the maps $C^{(r,s)}_{p,q}$ are given by
$$
C^{(r,s)}_{p,q} := \otimes^{r-1}_{s-1} \circ \tilde{P}^{(r,s)}_{p,q}
$$
Best Answer
There are two standard ways to index an element in a matrix.
First. If you note the matrix with uppercase letters from the beginning of the english alphabet, then you can use the lowercase version of the letter while indexing. For example the matrix is $A$ and the element in the $i$-th row and $j$-th column is $a_{ij}$. Or for $B$ you use $b_{ij}$. Etc. More at wikipedia about it.
Second. If the matrix is a most difficult expression, for example the Hadamard product if $A$ and $B$ matrices, and you note the Hadamard product with $\circ$. Then I prefer $[A \circ B]_{ij}$ notation. Some author also use the $\sideset{_i}{_j}{[A \circ B]}$ notation. It is also works for inverses of course: $[A^{-1}]_{ij}$.
Each way work with comma between indexes. As @Semiclassical wrote that, it is also a solution, that if the expression is very complex define an alternate variable.