$SE(3)$ represents the special Euclidean group which can be used to describe the position and orientation of a rigid body in 3D space. It can be defined as $$SE(3)=\{(p,R): R\in SO(3), p\in \mathbb{R}^3\}$$
When we say that we have an element $q \in SE(3)$ it means that $q$ is a pair of position $p$ and rotation matrix $R$.
To the Lie group $SE(3)$ can be associated the Lie algebra $\mathfrak{se}(3)$. I am writing a scientific paper and I was wondering if it is correct to say that an element $\tau \in \mathfrak{se}(3)$ can be seen as a pair of linear $v$ and angular $\omega$ velocities.
Thanks in advance
Yes, it is correct to say so. Although, the group structure of $SE (3)$ as you define is clearer if you put it in the language of transformations: an element $(p, R) \in SE (3)$ represents the transformation of rotation with the matrix $R$ followed by translation by $p$. Thus the transformation of the space is $x \mapsto Rx + p$. The composition of $(p, R)$ followed by $(q, S)$ is then represented by $(q + Sp, SR)$. This is the group structure. Moreover, for 'small' transformations, $p$ is small, and represents a linear velocity and $R = I + (R - I)$ is close to the identity $I$ and $R - I$ represents the angular velocity.