If $p>n$, then obviously this is true. For variables $x_1,\dots,x_n$, form the vector $\vec{v} = (x_1,\dots,x_n)^\dagger$, and consider the polynomial $F(x_1,\dots,x_n) = \text{det}(T_1\vec{v},\dots,T_n\vec{v})$. This is a homogeneous degree $n$ polynomial in the variables $x_1,\dots,x_n$ with coefficients in $\mathbb{F}_p$.
Now prove the following lemma: for every integer $n<p$, every polynomial $F(x_1,\dots,x_m)$ of degree $\leq n$ that vanishes identically on $\mathbb{F}_p^m$ is the zero polynomial. This is proved by induction on $m$. For $m=1$, this is just the statement that a nonzero, degree $n$ polynomial can have at most $n$ zeroes. Assuming the result for $m-1$, write $F(x_1,\dots,x_m)$ as $$F(x_1,\dots,x_n) = F_n(x_1,\dots,x_{m-1})x_m^n + \dots + F_0(x_1,\dots,x_{m-1})x_m^0.$$ For fixed values $(a_1,\dots,a_{m-1})\in \mathbb{F}_p^{m-1}$ the resulting polynomial $F(a_1,\dots,a_{m-1},x) \in \mathbb{F}_p[x]$ vanishes identically on $\mathbb{F}_p$ by hypothesis. Thus, by the $m=1$ case above, the coefficients $F_d(a_1,\dots,a_{m-1})$ are all zero. Thus by the induction hypothesis, every coefficient polynomial $F_d(x_1,\dots,x_{m-1})$ is the zero polynomial. Therefore $F(x_1,\dots,x_m)$ is the zero polynomial.