Let $M$ be a smooth manifold of even dimension $2n$. I would like to understand the equivalence of the following two standard definitions for an almost complex structure on $M$.
Denote by $TM$ the tangent bundle of $M$. Then the first definition I wish to consider is the following.
Definition 1 An almost complex structure on $M$ is a vector bundle morphism $J:TM\rightarrow TM$ satisfying $J^2=-id_{TM}$.
Now $TM$ is a real vector bundle of rank $2n$ over $M$ and is thus classified up to bundle isomorphism by the homotopy class of a map $\tau_M:M\rightarrow BGl(\mathbb{R}^{2n})$, where $BGl(\mathbb{R}^{2n})$ is the classifying space of the Lie group $Gl(\mathbb{R}^{2n})$. The complex general linear group $Gl(\mathbb{C}^n)$ also has a classifying space $BGl(\mathbb{C}^n)$, and the map $\mathbb{C}^n\rightarrow \mathbb{R}^{2n}$, $\underline z=\underline x+i\underline y\mapsto (\underline x,\underline y)$, corresponds to a homomorphism $r:Gl(\mathbb{C}^n)\rightarrow Gl(\mathbb{R}^{2n})$ given by
$$r(X)=r(A+iB)=\begin{pmatrix}A & -B\\B & A\end{pmatrix}$$
where $A$, $B$ are real matrices. This homomorphism subsequently induces a map of classifying spaces $Br:Gl(\mathbb{C}^n)\rightarrow BGl(\mathbb{R}^{2n})$ with homotopy fibre $Gl(\mathbb{R}^{2n})/Gl(\mathbb{C}^n)$.
Definition 2 An almost complex structure on $M$ is a lift through $Br$, up to homotopy, of the classifying map $\tau_M$. $$\begin{array}{ccccccccc} & & BGl(\mathbb{C}^n) \\ & \nearrow & \downarrow{Br}\\ M & \xrightarrow{\tau_M} & BGl(\mathbb{R}^{2n}). \end{array}$$
Remark that $BGl(\mathbb{R}^{2n})$ is modelled by the Grassmannian $Gr_{2n}(\mathbb{R}^{\infty})$ of $2n$-planes in $\mathbb{R}^\infty$, and a corresponding statement holds for $BGl(\mathbb{C}^{n})$. Since $M$ is finite dimensional the map $\tau_M$ will factor through the inclusion $Gr_{2n}(\mathbb{R}^{2N})\hookrightarrow Gr_{2n}(\mathbb{R}^{\infty})$ for some $N\geq n$, and $Gr_{2n}(\mathbb{R}^{2N})$ may be given the structure of a smooth manifold and $\tau_M$ represented by a smooth map. Similarly $Gr_{N}(\mathbb{C}^{N})$ carries the structure of a complex manifold and $Br|$ may be chosen to be a smooth map between these spaces.
Now my question: How are these two definition equivalent?