I have trouble to properly understand the definition of a riemannian metric.
Let $M$ be a smooth manifold. We have the definition of tensor fields as the smooth sections of
$T^{r,s} M=\underbrace{TM \otimes ... \otimes TM}_{r} \otimes \underbrace{T^{*}M \otimes...\otimes T^{*}M}_s$
Then we have defined, that a pseudo-Riemannian metric is a $(0,2)$ tensor field $g$ on $M$, where $g_p$ is a scalar product for each $p \in M$
That means, that $g(p)$ is a bilinear form $T_pM \times T_p M \rightarrow \mathbb{R}$, right?
so if $g$ is a $(0,2)$ tensor field, then $g:M \rightarrow T^{*}M \otimes T^{*}M$, so $g_p \in T_pM ^{*}\otimes T_pM^{*}, g_p= \sum\limits_{i,j=1}^n c_{ij} (e_j \otimes e_j)$, where $c_{ij} \in \mathbb{R}$ and $(e_i)$ is a basis of $T_pM^{*}$. Now let $v,w \in T_pM$, then $g_p(v,w)= \sum\limits_{i,j=1}^n c_{ij} (e_j(v) \otimes e_j(w)) \in \mathbb{R}$
Is my interpretation correct? It's not that I have a concrete question, it's just that my knowledge also of tensors is not very deep so I'm not quite sure about it.