  Over an algebraically closed base field $k$ of characteristic 2, the ring
$R^G$ of invariants is studied, $G$ being the orthogonal group O(n) or the
special orthogonal group SO(n) and acting naturally on the coordinate ring $R$
of the $m$-fold direct sum $k^n \oplus...\oplus k^n$ of the standard vector
representation. It is proved for O(n) $(n\geq 2)$ and for SO(n) $(n\geq 3)$
that there exist $m$--linear invariants with $m$ arbitrarily large that are
indecomposable (i. e., not expressible as polynomials in invariants of lower
degree). In fact, they are explicitly constructed for all possible values of
$m$. Indecomposability of corresponding invariants over $\mathbb Z$ immediately
follows. The constructions rely on analysing the Pfaffian of the skew-symmetric
matrix whose entries above the diagonal are the scalar products of the vector
variables.
