For a concise introduction to RKHS, you could have a look at sections 2.3 and 2.4 of Gaussian Processes and Kernel Methods: A Review on Connections and Equivalences by Kanagawa et al. (2018).
In particular, they give a characterisation of the RKHS associated to a shift-invariant kernel on $\mathbb{R}^d$. In this case, the inner product is linked to the Fourier transform of the kernel:
Theorem 2Theorem 2.4 (Kanagawa et al.4)
Let $k(x, y) = \Phi(x - y)$ be a shift-invariant kernel on $\mathbb{R}^d$. Also and assume that $k \in C(\mathbb{R}^d) \cap L_1(\mathbb{R}^d)$.
Then the RKHS of k is given by
$$ \mathcal{H}_k = \Big\lbrace f \in C(\mathbb{R}^d) \cap L_1(\mathbb{R}^d), ~||f||_{\mathcal{H}_k} < \infty \Big\rbrace $$
$$ \mathcal{H}_k = \Big\lbrace f \in C(\mathbb{R}^d) \cap L_1(\mathbb{R}^d), ~||f||_{\mathcal{H}_k} < \infty \Big\rbrace $$ Where Where the inner product is given by $$ \langle f, g \rangle_{\mathcal{H}_k} = \frac{1}{(2\pi)^{d/2}}\int\frac{\mathcal{F}[f](\omega)\overline{\mathcal{F}[g](\omega)}}{\mathcal{F}[\Phi](\omega)}d\omega $$
$$ \langle f, g \rangle_{\mathcal{H}_k} = \frac{1}{(2\pi)^{d/2}}\int\frac{\mathcal{F}[f](\omega)\overline{\mathcal{F}[g](\omega)}}{\mathcal{F}[\Phi](\omega)}d\omega $$
If you're working with more exotic spaces or kernels (complex or bounded spaces, periodic kernels, ...) you could have a look at the book by Berlinet and Thomas-Agnan Reproducing Kernel Hilbert Spaces in Probability and Statistics.
Chapter 7.4 gives a list of RKHSs together with associated kernel and analytical expression for the inner product.
Finally, Chapter 6.2 mentions a numerical scheme for computing the RKHS norm in a special case. Maybe this could be interessting to you.