Cross product between vecter and tensor

[MatCond]

Homework Statement

Just wanted to ask what's the definition of the cross product between a vector and a range two tensor

The Attempt at a Solution

$$(x \times \hat{T})_{i\beta}=\epsilon_{ijk} x_j T_{k\beta}$$

 

[yunde shi]

based on wikiversity,
http://en.wikiversity.org/wiki/Introduction_to_Elasticity/Tensors
it looks like
$$(\hat{T} \times x)_{i\beta}=\epsilon_{jk\beta} T_{ij} x_k$$
then
$$(x \times \hat{T})_{i\beta}=\epsilon_{ijk} x_j T_{k\beta}$$
?

 

[yunde shi]

if those were true, then is the following correct ?

$$<br /> -(\hat{T}^{T} \times x) = (x \times \hat{T})^{T}<br />$$