The geometric definition of the cross product $a\times b$ of two vectors (arrows) $a$ and $b$ seems unambiguous, in particular its direction is uniquely determined by the right hand rule.
So why is the cross product called a pseudo vector (or pseudo form) whose direction depends on the orientation of the vector space? There must be a connection between the geometric definition and the formula $a\times b=*(a\wedge b)$ I found, where the dependency on the orientation is explicit, musn't it?
In $\mathbb{R}^3$ with the canonical basis $B=(e_1,e_2,e_3)$, there exist two different orientations. The classical one is saying $B$ is positively oriented, the other is saying $B$ is negatively oriented.
The cross product of two linearly independant vectors $u$ and $v$ is defined to be the unique vector $w$, orthogonal to $u$ and $v$, such that $\|w\| = \|u\|\|v\||\sin(u,v)|$ and $(u,v,w)$ is a positively oriented basis. Positively oriented means that $\det_B(u,v,w) >0$ if $B$ is positively oriented, $<0$ otherwise.
The second point is where the orientation of $\mathbb{R}^3$ matters. If you chose $B$ to be positively oriented, then $e_1 \times e_2 = e_3$ (right hand rule). If you say that $B$ is negatively oriented, then $e_1\times e_2 = -e_3$. In general, the change of orientation induces a multiplication by $(-1)$ on the cross product.
In a general orientable riemannian manifold, a metric $g$ induces the Hodge operator $\star$, which can define similar things. It closely depend on the volume form induced by the metric $g$ and thus, on the orientation of the manifold.