The extension to $g>3$ of Ichikawa work leads to vector-valued modular forms. The main steps are the following.
For each fixed positive integers
$g,n$, we define  $$M_n(g)=M_n:={g+n-1\choose n}\ ,\quad
N_n(g)=N_n:=(2n-1)(g-1)\ ,\quad K_n:=M_n-N_n$$

Let $\{\phi^n_i\}_{1\le i\le N_n}$ be a basis of
$H^0(K_C^n)$, $n\geq2$. For any points $p,x_1,\ldots,x_{N_n}\in C$,
the Mumford form is, up to a universal constant
$$
\mu_{g,n}={\kappa[\omega]^{(2n-1)^2}\over
\kappa[\phi^n]}{\phi^n_1\wedge\cdots\wedge\phi^n_{N_n}\over
(\omega_1\wedge\cdots\wedge\omega_g)^{c_n}}
$$
where $\omega_1,\ldots,\omega_g$ is the standard (normalized) basis of $H^0(K_C)$. $\kappa[\omega]$ is a constant that depends only on the choice of the homological basis whereas $\kappa[\phi^n]$ also depends on the choice of the basis (see Prop.1.2). In the case $n=2$ and $n<4$ you may choose a natural basis for $H^0(K_C^2)$: $\Sym^2 H^0(K_C)$, and you get
for $g=2$ $${\kappa[\omega]^{9}\over
\kappa[\omega^{(2)}]} ={1\over \pi^{12}\chi_{5}^2(\tau)}\ ,$$
whereas for $g=3$
$${\kappa[\omega]^{9}\over
\kappa[\omega^{(2)}]} ={1\over 2^6\pi^{18}\chi_{18}^{1/2}(\tau)}\ . $$
For $g>3$ you have $g(g+1)/2-(3g-3)>0$, so apparently there is no a natural extension. Nevertheless you may continue to take $3g-3$ elements of $\Sym^2 H^0(K_C)$ (or more generally $N_n:=(2n-1)(g-1)$ elements of $\Sym^n H^0(K_C)$. What you get is

1. ${\kappa[\phi^n]\over\kappa[\omega^{(n)}]^{(2n-1)^2}}$, that essentially correspond to what we denoted by $$[i_{N_n+1},\ldots,i_{M_n}|\tau] \ , $$ are vector-valued modular forms of weight
$$d_n:=6n^2-6n+1-{g+n-1\choose n-1} \ .$$


2. For each integer $n\geq 2$ and for all $i_{2},\ldots,i_{K_n}\in\{1,\ldots,M_n\}$ we have
$$\sum_{i=1}^{M_n}[i,i_{2},\ldots,i_{K_n}|\tau]\omega^{(n)}_{i}(x)
=0\ .$$ In particular, for $n=2$ these are all the quadrics characterizie the canonical curve in projective space.

3. As a first step that such vector-valued are interesting objects one notes that at $g=4$
$$\det S_4(\tau)=d\chi_{68}(\tau)^{1/2}\ ,$$
where $S_4$ is the derivative with respect to the $Z_{ij}\in {\cal H_4}$ evaluated on the Schottky locus of the Schottky-Igusa form and $\chi_{68}$ is the $g=4$ Thetanullwerte. It is nice that $S_4$ is proportional to $[i|\tau]$.