How to compute the ring of invariants of SO_3(k) acting on a polynomial ring Let $k$ be a field and let $A$ be the polynomial ring over $k$ in $3n$ variables: $A = k[X_{ij} \vert i=1,2,3 \quad j=1,2,\cdots,n]$.
${\rm SO}_3(k)$ acts on $A$ in the following way: Given  $g \in {\rm SO}_3(k)$, we define:
$$g(X_{ij})=g_{ik}X_{kj}$$ 
with respect to the summation convention.
Can the ring of invariants of this action be expressed in terms of generators and relations?  I get the feeling that this is a standard exercise in invariant theory, but am not sure where to look. 
 A: This is addressed by the classical invariant theory, but the answer is more complicated than for general linear or orthogonal groups (in particular, not all minimal generators are quadratic). Let $k$ be a field of characteristic 0. The group $G=SO_m$ acts on $m\times n$ matrices by the left multiplication and this induces a $G$-action on $A=k[X_{ij}].$ Let us view the variables as the entries of the $m\times n$ generic matrix over $k.$ Then the algebra of invariants $A^G$ is generated by:
1 Scalar products of the columns of the matrix $X.$
2 Order $m$ minors of the matrix $X.$
This is the First Fundamental Theorem (FFT) of classical invariant theory for $SO_m.$ In fact, the elements of the first type generate $O_m$-invariants and the elements of the second type generate $SL_m$-invariants ($SO_m=O_m\cap SL_m$).
Moreover, all relations between these generators are also known (the Second Fundamental Theorem, SFT) and there is a good description of a standard monomial basis of $A^G.$ If I am not mistaken, the last part is due to Laskshmibai and coauthors. A comprehensive modern reference is

Laskshmibai and Raghavan, Standard monomial theory. Invariant theoretic approach. Encyclopaedia of Mathematical Sciences, vol 137 (Invariant Theory and Algebraic Transformation Groups VIII), Springer. 

A: H. Weyl, The classical groups chapter V
