Consider a form of degree $r$ in $n$, that is, a homogeneous polynomial
$$f(x_1, \ldots, x_n)=\sum_{i_1+\ldots i_n=r}\alpha_{i_1 ... i_n}x_1^{i_1} ... x_n^{i_n} $$
After the linear change of variables $$\delta : x^\prime_1\rightarrow\sum_{j=1}^n\beta_{ij}x_{j}\;\;\;\;\;\;\;\;\;\;1\leq i\leq n,$$
where $\beta_{ij}$ are real or complex numbers, we obtain a new form
$$f^\prime(x^\prime_1, \ldots, x^\prime_n)=\sum_{i_1+\ldots i_n=r}\alpha^\prime_{i_1 ... i_n}{x^\prime_1}^{i_1} ... {x^\prime_n}^{i_n} $$
An invariant of the form $f$ is a polynomial function of the coefficients $\alpha_{i_1 ... i_n}$ that changes only by a factor equal to a power of the determinant of the linear transformation $\delta$ if one replaces the coefficients $\alpha_{i_1 ... i_n}$ of the given base form by the corresponding coefficients $\alpha^\prime_{i_1 ... i_n}$of the linearly transformed form.
This is the definition of invariants given by Hilbert in "Theory of Algebraic Invariants".
My question is: Consider the determinant of the matrix $A=[x_{ij}]_{n×n}$ as a polynomial in the ring $R[x_{11},…,x_{nn}]$. Since the determinant is a homogeneous polynomial it is a form. What are the invariants of the determinant form?